#509537

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

517 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)(5,6,9,12)(7,8,11,10)(13,14)(15,21)(16,22)(17,20)(18,23)(19,24)(25,37)(26,36)(27,33)(28,38)(29,34)(30,40)(31,39)(32,41)(35,42)(43,44);(5,7,9,11)(6,10,12,8)(14,20)(15,25)(16,19)(18,30)(21,35)(23,38)(26,43)(27,42)(28,29)(31,32)(33,37)(34,40)(36,39)(41,44);(2,4)(5,8,9,10)(6,11,12,7)(13,15,17,27)(14,21,20,33)(16,26,19,32)(18,31,29,43)(22,36,24,41)(23,39,34,44)(25,28,42,30)(35,40,37,38);(2,4)(6,12)(7,11)(13,16,17,19)(14,22,20,24)(15,26,27,32)(18,28,29,30)(21,36,33,41)(23,38,34,40)(25,31,42,43)(35,44,37,39)

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416733);