#509911

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

571 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)(15,23)(16,20)(17,21)(18,22)(19,25)(24,32)(26,41)(27,40)(28,35)(29,36)(30,39)(31,34)(33,43)(37,42)(38,44);(5,7,9,6)(8,11,12,10)(13,15,16,25)(14,19,20,23)(17,24,28,42)(18,31,29,27)(21,37,35,32)(22,40,36,34)(26,44,43,30)(33,38,41,39);(2,4)(5,8,9,12)(6,10,7,11)(14,20)(15,24)(17,30)(18,29)(19,32)(21,38)(23,37)(25,42)(26,27)(28,44)(31,43)(33,40)(34,41)(35,39);(2,4)(6,7)(10,11)(15,25)(19,23)(24,42)(26,43)(27,31)(32,37)(33,41)(34,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=ei,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, 417107);