#497158

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

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=dei,c^b=cei,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, 404354);