#496245

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1782 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,16)(13,26,27,17)(14,18)(15,21,29,36)(20,33)(22,34)(23,30)(24,38)(25,40,37,32)(28,35,31,39);(1,3)(2,5)(4,8)(6,7)(12,23)(16,30)(20,24)(25,37)(28,31)(32,40)(33,38)(35,39);(3,8)(5,7)(9,11,14,22)(10,15,18,29)(12,24,23,20)(13,21,27,36)(16,31,30,28)(17,19,26,34)(25,39,37,35)(32,38,40,33);(2,6)(5,7)(9,12,14,23)(10,16,18,30)(11,20,22,24)(13,25,27,37)(15,28,29,31)(17,32,26,40)(19,33,34,38)(21,35,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403441);