#95297

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

556 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)(3,5)(4,7)(6,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24)(25,26)(27,30,34,37)(28,31)(29,32)(33,36,39,40)(35,38);(2,5)(3,8)(6,7)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)(25,27)(26,30,32,37)(28,33)(29,34)(31,36,38,40)(35,39);(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,36,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2493);