#496079

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator, socle, prime radical	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci

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, 403275);