#362499

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269695);