#493169

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

748 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400365);