#480078

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1060 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)(3,7,8,5)(4,6)(9,10,13,17,14,18,27,31,15,19,12,24,25,32,26,37)(11,16,22,28,23,30,36,38,21,29,20,33,34,39,35,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,28)(18,30)(19,29)(24,33)(25,34)(26,35)(27,36)(31,38)(32,39)(37,40);(9,12,14,26,15,13,25,27)(10,17,18,31,19,24,32,37)(11,20,23,35,21,22,34,36)(16,28,30,38,29,33,39,40);(11,21)(16,29)(20,22)(23,34)(28,33)(30,39)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=h,h^2=i,b^a=be,c^a=ci,e^a=eg,d^b=di,f^b=fg

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, 387274);