#361026

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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

1262 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)(11,21,24,31,25,36,23,16)(12,17,27,34,28,35,29,33)(22,37,38,40,39,32,26,30);(9,11)(10,16)(12,26)(13,24)(14,23)(15,25)(17,32)(18,36)(19,21)(20,31)(22,27)(28,38)(29,39)(30,34)(33,40)(35,37);(2,5)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,33)(19,34)(20,35)(21,37)(23,26)(24,38)(25,39)(31,40)(32,36);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,24)(22,30)(23,36)(25,31)(26,32)(27,34)(28,35)(29,33)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,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^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268222);