#480012

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1672 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:

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

Polycyclic presentation:

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

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, 387208);