#6826819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1516 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)(9,10,13,15)(11,16,12,14)(17,18)(19,24)(20,25)(21,23)(22,26)(27,40)(28,39)(29,35)(30,45)(31,42)(32,36)(33,41)(34,37)(38,44)(43,48)(46,47);(9,10)(11,14)(12,16)(13,15)(18,23)(19,27)(20,30)(22,34)(24,38)(25,41)(29,44)(32,33)(35,40)(36,45)(39,42)(43,46);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,27,21,29,34,44)(18,24,26,40,23,35,37,38)(20,31,33,46,32,28,30,43)(25,42,41,47,36,39,45,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(18,23)(20,32)(24,35)(26,37)(28,31)(30,33)(38,40)(43,46);(9,12,13,11)(10,14,15,16)(17,20,22,33,21,32,34,30)(18,25,26,41,23,36,37,45)(19,28,27,43,29,31,44,46)(24,39,40,48,35,42,38,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6734015);