#509435

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

683 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,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, 416631);