#500458

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

795 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)(3,7)(4,5)(6,8)(9,10,14,19)(11,15,24,30)(12,16,25,17)(13,18,27,35)(20,34,37,40)(21,31,23,29)(22,28,36,39)(26,32,38,33);(2,5)(6,7)(9,11)(10,15)(12,21)(13,20)(14,24)(16,31)(17,29)(18,34)(19,30)(22,26)(23,25)(27,37)(28,32)(33,39)(35,40)(36,38);(10,16)(11,20)(12,25)(13,26)(15,28)(17,19)(18,35)(21,36)(22,23)(24,37)(27,38)(29,40)(30,39)(31,34);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 407654);