#95230

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bd,c^a=cg,d^a=dg,f^a=fghi,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2426);