#496753

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×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

571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,28,33,29,34,41,43,27,35,39,44,40,45,47,48);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(10,13)(14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,35)(25,33)(26,34)(30,41)(31,39)(32,40)(36,44)(37,45)(38,43)(42,47)(46,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(25,36)(26,37)(28,39)(29,40)(38,46)(41,47)

Polycyclic presentation:

2^9|a^2=g,d^2=f,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, 403949);