#72234381

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#2263	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ #11408 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,6,5)(3,7,8,4)(9,10)(11,17)(12,25)(13,31)(14,36)(15,38)(16,21)(18,28)(19,35)(20,34)(22,48)(23,52)(24,47)(26,54)(27,45)(29,63)(30,64)(32,42)(33,65)(37,66)(39,50)(40,53)(41,60)(43,57)(44,70)(46,49)(51,68)(55,62)(56,61)(58,72)(59,67)(69,71);(1,3,5,4,6,8,2,7)(12,22,26,30)(13,32)(18,41,43,39)(19,45)(23,33)(25,48,54,64)(27,35)(28,60,57,50)(29,51,58,61)(31,42)(40,46)(44,67,69,55)(49,53)(52,65)(56,63,68,72)(59,71,62,70);(1,4,5,8,6,7,2,3)(9,11,16,15)(10,17,21,38)(12,26)(13,33,32,23)(14,24,37,34)(19,46,45,40)(20,36,47,66)(22,30)(25,54)(27,53,35,49)(29,58)(31,65,42,52)(48,64)(51,61)(56,68)(63,72);(9,12,27,59,14,29,33,60,11,22,49,70,24,51,13,28,16,26,35,62,37,58,23,50,15,30,53,71,34,61,32,57)(10,18,42,68,20,44,40,48,38,41,52,63,66,67,19,25,21,43,31,56,47,69,46,64,17,39,65,72,36,55,45,54)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=fj,d^2=egj,e^2=gi,f^2=h,g^2=ij,i^2=j,c^b=ch,d^a=de,d^b=dij,d^c=dij,e^a=eg,e^b=ej,e^c=ej,f^d=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234391, #72322300, #72322334
maximal normal subgroup of	#72234391, #72322300, #72322334

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 4, F.5*F.7*F.10);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);