#494769

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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)(7,10)(8,9)(11,12)(13,14,18,23,19,24,33,40,17,22,31,38,32,39,34,44)(15,20,26,16,27,29,42,30,25,35,41,28,21,36,37,43);(6,9)(10,12)(13,15,18,26,19,27,33,42,17,25,31,41,32,21,34,37)(14,20,23,16,24,29,40,30,22,35,38,28,39,36,44,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,28)(18,29)(19,30)(20,34)(22,27)(23,37)(25,39)(26,40)(31,36)(32,43)(33,35)(38,42)(41,44);(14,22)(16,28)(20,35)(23,38)(24,39)(29,36)(30,43)(40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401965);