#512283

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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)(15,19)(16,20)(17,30)(18,23)(21,27)(22,28)(24,31)(25,40)(26,34)(29,36)(32,37)(33,38)(35,41)(39,43)(42,44);(1,2,4,3)(5,7,8,11)(6,9,10,12);(6,10)(9,12)(13,15,18,26)(14,19,23,34)(16,24,29,39)(17,25,28,38)(20,31,36,43)(21,32,35,42)(22,33,30,40)(27,37,41,44);(1,3,4,2)(14,20)(16,27)(17,28)(19,31)(21,22)(23,36)(24,37)(25,38)(29,41)(30,35)(32,33)(34,43)(39,44)(40,42)

Polycyclic presentation:

2^9|b^2=ef,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, 419479);