#501315

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408511);