#502873

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

892 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)(4,5)(6,8)(9,10)(11,16)(12,25)(13,29)(14,32)(15,19)(17,20)(18,27)(21,36)(22,34)(23,39)(24,40)(26,37)(28,31)(30,38)(33,35);(2,5)(6,7)(9,11)(10,16)(12,26)(13,30)(14,24)(15,22)(17,33)(18,21)(19,34)(20,35)(23,31)(25,37)(27,36)(28,39)(29,38)(32,40);(9,12,27,19,14,28,31,32,15,18,25,10,17,29,13,20)(11,21,37,34,24,38,30,40,22,26,36,16,33,39,23,35);(1,3,4,8)(2,6,5,7)(11,22)(16,34)(21,26)(23,30)(24,33)(35,40)(36,37)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410069);