#495662

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

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

522 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,27,22,28,34,43,23,26,35,41,36,42,46,48)(19,24,29,37,30,38,44,20,25,32,39,33,40,45,47,31);(2,5)(6,8)(17,19,21,29,22,30,34,44,23,25,35,39,36,40,46,47)(18,24,27,37,28,38,43,20,26,32,41,33,42,45,48,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,25)(19,26)(21,32)(22,33)(23,31)(24,35)(27,39)(28,40)(29,41)(30,42)(34,45)(36,37)(38,46)(43,47)(44,48);(9,11,12,15)(10,14,13,16)(18,26)(20,31)(24,32)(27,41)(28,42)(33,37)(38,45)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^c=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, 402858);