#509801

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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)(13,14,20,35,15,26,33,38,17,24,18,25,19,21,28,23)(16,29,43,41,34,44,32,22,30,39,31,37,27,42,36,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,21,24,26)(16,27,30,34)(18,28,20,33)(22,37,40,41)(23,25,38,35)(29,44,39,42)(31,32,43,36);(2,4)(13,16,17,30)(14,22,24,40)(15,27,19,34)(18,32,20,36)(21,37,26,41)(23,39,38,29)(25,42,35,44)(28,43,33,31);(6,10)(7,12)(14,23)(15,19)(18,33)(20,28)(21,35)(22,39)(24,38)(25,26)(27,34)(29,40)(31,32)(36,43)(37,44)(41,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 416997);