#362362

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

1261 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)(9,10,13,19,15,20,14,18)(11,21,24,32,25,31,23,16)(12,17,27,35,28,36,29,34)(22,33,26,39,38,40,37,30);(9,11,14,23,15,25,13,24)(10,16,18,31,20,32,19,21)(12,26,27,38,28,37,29,22)(17,33,35,39,36,40,34,30);(2,5)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,35)(20,36)(21,33)(23,37)(24,26)(25,38)(31,40)(32,39);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,24)(22,30)(23,31)(25,32)(26,33)(27,35)(28,36)(29,34)(37,40)(38,39)

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,e^b=ei,g^c=gi,e^d=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, 269558);