#493876

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1687 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)(9,10,13,18,14,19,26,33)(11,15,22,30,20,28,35,39)(12,17,24,32,23,16,25,31)(21,29,36,40,34,27,37,38);(2,5)(6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,29)(18,30)(19,28)(23,34)(24,36)(25,37)(26,35)(31,38)(32,40)(33,39);(1,3,4,7)(2,6,5,8)(10,16)(12,23)(15,27)(17,19)(18,32)(21,34)(24,25)(28,29)(30,40)(31,33)(36,37)(38,39);(11,20)(15,28)(21,34)(22,35)(27,29)(30,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=g,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fi,g^c=ghi,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, 401072);