#494004

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1107 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,25,26,17)(14,18)(16,23,30,37)(19,31)(20,28)(21,35,34,29)(24,36)(27,33,38,40)(32,39);(2,5)(6,8)(9,11,14,20)(10,15,18,28)(12,19,24,32)(13,21,26,34)(16,27,30,38)(17,29,25,35)(22,31,36,39)(23,33,37,40);(9,12,14,24)(10,16,18,30)(11,19,20,32)(13,23,26,37)(15,27,28,38)(17,22,25,36)(21,33,34,40)(29,31,35,39);(1,3,4,7)(2,6,5,8)(11,20)(15,28)(19,32)(21,34)(27,38)(29,35)(31,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=e,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,g^c=gh,g^f=gi

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, 401200);