#480120

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
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	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,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, 387316);