#509865

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
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

531 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)(7,8)(9,11)(10,12)(13,14,16,21,28,36,40,35,18,23,29,34,17,22,27,20)(15,19,24,31,38,43,44,42,26,33,39,41,25,32,37,30);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,26)(14,19,23,33)(16,24,29,39)(17,25,28,38)(20,30,35,42)(21,31,34,41)(22,32,36,43)(27,37,40,44);(2,4)(5,8,9,10)(6,7,11,12)(14,20)(16,27)(17,28)(19,30)(21,22)(23,35)(24,37)(25,38)(29,40)(31,32)(33,42)(34,36)(39,44)(41,43);(6,11)(7,12)(14,20)(16,27)(17,28)(19,30)(21,22)(23,35)(24,37)(25,38)(29,40)(31,32)(33,42)(34,36)(39,44)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,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, 417061);