#495448

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
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

634 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)(13,14,16,21,17,22,26,34,18,20,27,32,28,33,38,42)(15,19,23,30,24,31,35,41,25,29,36,39,37,40,43,44);(2,4)(13,15,16,23,17,24,26,35,18,25,27,36,28,37,38,43)(14,19,21,30,22,31,34,41,20,29,32,39,33,40,42,44);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,20)(19,29)(21,32)(22,33)(30,39)(31,40)(34,42)(41,44)

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^a=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, 402644);