#72314576

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33582	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20437 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,4,3)(5,6)(7,15)(8,13)(9,14)(10,11)(12,16)(17,26)(18,32)(19,33)(20,31)(21,30)(22,24)(23,25)(27,35)(28,34)(29,36);(1,3,4,2)(7,16)(12,15)(17,34)(18,35)(19,36)(26,28)(27,32)(29,33);(6,11)(7,17,16,34)(8,20)(9,23)(12,26,15,28)(13,30)(18,36,35,19)(21,22)(24,31)(27,29,32,33);(2,3)(7,18)(8,21)(9,23)(12,27)(13,30)(14,25)(15,32)(16,35)(17,36)(19,34)(20,22)(24,31)(26,29)(28,33);(2,3)(5,7,8,17,9,19,20,35,10,16,22,34,23,36,21,18)(6,12,13,28,14,29,31,32,11,15,24,26,25,33,30,27)

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=f,e^2=g,g^2=i,i^2=j,c^a=cj,d^a=dh,d^b=dh,e^a=ehj,e^b=ehj,e^c=eg,e^d=egij,f^e=fj,g^c=gi,g^d=gij,i^c=ij,i^d=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314594, #72315346, D₄×#33471
maximal normal subgroup of	#72314594, #72315346, D₄×#33471

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