#496076

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

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

473 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,20,32,34,23,27,36,43,37,35,25,41)(19,24,30,40,31,33,39,47,28,38,44,48,45,29,42,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,28)(25,42)(26,40)(27,38)(29,35)(32,39)(34,47)(36,44)(37,45)(41,46)(43,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,28)(24,38)(29,33)(30,44)(31,45)(39,42)(40,48)(46,47);(17,20,21,34,22,27,32,43,23,35,36,41,37,18,25,26)(19,29,30,46,31,24,39,40,28,33,44,47,45,38,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,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

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, 403272);