#498652

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, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

428 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,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, 405848);