#234591

order	512
nilpotency class	3
isoclinism	isoclinic class of #232273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

4411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20);(6,10)(7,8)(11,12)(17,20);(2,4)(13,15)(14,17)(16,19)(18,20);(5,7,9,8)(6,11,12,10)(15,19)(17,20)

Polycyclic presentation:

2^9|d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210545, #72210583, GL(2,ℤ/8), …
maximal normal subgroup of	#72210545, #72210583, #72213195, …
maximal quotient of	#72217661, #72217697, #72217702, …
automorphism group of	#322
outer automorphism group of	#1311, C₈×C₃⋊C₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141787);