#497154

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
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

494 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)(12,16)(13,15)(17,18,22,27,23,19,28,31,21,26,36,43,37,30,24,38)(20,25,34,42,35,29,40,45,33,41,44,48,47,32,39,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,31,23,26,28,43,21,30,36,38,37,18,24,27)(20,32,34,46,35,25,40,42,33,29,44,45,47,41,39,48);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,35)(24,39)(26,41)(27,42)(28,40)(30,32)(31,45)(36,44)(37,47)(38,46)(43,48);(2,6)(5,7)(18,26)(19,30)(20,33)(27,43)(31,38)(34,44)(35,47)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,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, 404350);