#95328

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

488 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)(3,7)(4,6)(5,8)(9,10,12,18,14,16,13,17)(11,19,21,15,20,23,22,24)(25,26)(27,30,34,38)(28,31)(29,32)(33,37,35,39)(36,40);(2,6)(3,4)(7,8)(9,11,13,22,14,20,12,21)(10,15,18,23,16,24,17,19)(25,27)(26,30,32,38)(28,35)(29,34)(31,39,40,37)(33,36);(1,3,5,4)(2,7,8,6)(10,16)(11,20)(17,18)(21,22)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,37,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239616
maximal normal subgroup of	#72239616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2524);