#480359

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂×#525	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1038 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387555);