#72231960

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂⁴	#346
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

745 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=e,e^2=g,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=cj,d^a=dij,d^b=dj,e^a=ej,f^d=fj

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231961, #72231969
maximal normal subgroup of	#72231961, #72231969

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.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);