#501797

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,18)(15,20,25,34)(16,21,29,27)(17,22,33,32)(23,35,40,39)(24,30,37,42)(26,36,43,41)(28,31,38,44);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,26)(17,30)(18,34)(19,25)(21,36)(22,37)(23,28)(24,32)(27,41)(29,43)(31,35)(33,42)(38,40)(39,44);(6,10)(9,12)(13,16,19,29)(14,21,18,27)(15,23,25,40)(17,31,33,44)(20,35,34,39)(22,38,32,28)(24,41,37,36)(26,42,43,30);(2,4)(13,14,19,18)(15,20,25,34)(16,27,29,21)(17,22,33,32)(23,39,40,35)(24,30,37,42)(26,41,43,36)(28,44,38,31)

Polycyclic presentation:

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

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, 408993);