#509670

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

588 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,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, 416866);