#126906

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,11,12,9)(8,10)(13,14,17,22,18,23,32,40,19,24,33,41,16,21,30,38)(15,25,28,42,27,35,43,39,26,37,31,44,29,20,34,36);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,29)(17,31)(18,27)(19,26)(21,37)(22,39)(23,25)(24,35)(28,33)(30,43)(32,34)(36,41)(38,42)(40,44);(15,26)(20,35)(25,37)(27,29)(28,31)(34,43)(36,39)(42,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,c^b=ci,d^b=di,e^b=egi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34102);