#494014

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	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1064 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,26,27,17)(14,18)(16,24,32,40)(19,33)(20,29)(21,36,35,30)(23,37)(25,38)(28,34,31,39);(2,5)(6,8)(9,11,14,20)(10,15,18,29)(12,23,25,19)(13,21,27,35)(16,31,32,28)(17,30,26,36)(22,37,38,33)(24,39,40,34);(9,12,14,25)(10,16,18,32)(11,19,20,23)(13,24,27,40)(15,28,29,31)(17,22,26,38)(21,34,35,39)(30,33,36,37);(1,3,4,7)(2,6,5,8)(11,20)(15,29)(19,23)(21,35)(28,31)(30,36)(33,37)(34,39)

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,c^b=ci,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, 401210);