#481056

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂²×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388252);