#126531

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₄²×C₈
socle, prime radical	C₂³	C₂×C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bi,c^b=cg

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210241, C₂×#126531, #72210416, …
maximal normal subgroup of	#72210241, C₂×#126531, #72210416, …
maximal quotient of	C₂×#126531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33727);