#496024

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	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=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, 403220);