#495457

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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

602 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)(13,14,18,23,19,15,24,27,17,22,31,37,32,26,20,33)(16,21,29,36,30,25,38,40,28,35,41,44,42,39,34,43);(2,4)(13,15,18,27,19,22,24,37,17,26,31,33,32,14,20,23)(16,25,29,40,30,35,38,44,28,39,41,43,42,21,34,36);(6,9)(10,12)(13,16)(14,21)(15,25)(17,28)(18,29)(19,30)(20,34)(22,35)(23,36)(24,38)(26,39)(27,40)(31,41)(32,42)(33,43)(37,44);(5,7,8,11)(6,10,9,12)(16,28)(21,35)(25,39)(29,41)(30,42)(34,38)(36,44)(40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,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, 402653);