#512285

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efi,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,g^d=gh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419481);