#362363

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1251 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)(4,5)(7,8)(9,10,16,19,15,13,14,20)(11,21,23,17,24,33,25,31)(12,26,30,36,29,18,28,35)(22,34,27,40,38,37,39,32);(9,11,14,25,15,24,16,23)(10,17,20,21,13,31,19,33)(12,27,30,38,29,39,28,22)(18,34,35,40,26,37,36,32);(2,5)(6,8)(9,12)(10,18)(11,22)(13,26)(14,28)(15,29)(16,30)(17,32)(19,35)(20,36)(21,37)(23,27)(24,38)(25,39)(31,40)(33,34);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,17,24,31)(12,26,29,18)(14,19,16,20)(21,23,33,25)(22,32,38,40)(27,34,39,37)(28,35,30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 269559);