#496023

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

592 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)(13,14,18,22,19,23,31,36,17,16,21,28,30,29,35,41)(15,20,25,32,26,33,38,42,24,27,34,39,37,40,43,44);(1,2,4,3)(6,9)(10,12)(13,15,17,24)(14,20,16,27)(18,25,21,34)(19,26,30,37)(22,32,28,39)(23,33,29,40)(31,38,35,43)(36,42,41,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,16)(15,24)(22,28)(23,29)(25,34)(26,37)(36,41)(38,43);(13,16,18,28,19,29,31,41,17,14,21,22,30,23,35,36)(15,20,25,32,26,33,38,42,24,27,34,39,37,40,43,44)

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,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, 403219);