#501789

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

705 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);(2,4)(5,7,8,11)(6,9,10,12)(13,14)(15,24)(16,21)(17,30)(18,34)(19,23)(20,26)(22,32)(25,38)(27,44)(28,41)(29,39)(31,37)(33,40)(35,42)(36,43);(6,10)(9,12)(13,15,19,28)(14,20,23,36)(16,27,18,25)(17,31,33,42)(21,37,34,35)(22,38,39,44)(24,40,41,30)(26,29,43,32);(2,4)(13,16,19,18)(14,21,23,34)(15,25,28,27)(17,29,33,32)(20,35,36,37)(22,30,39,40)(24,38,41,44)(26,42,43,31)

Polycyclic presentation:

2^9|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, 408985);