#496715

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
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	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403911);