#6802671

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25727	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709867);