#6828908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,23)(22,26)(27,39)(28,34)(29,38)(30,44)(31,36)(32,35)(33,40)(37,41)(42,48)(43,46)(45,47);(9,11)(10,13)(12,16)(14,15)(18,23)(19,27)(20,30)(22,31)(24,37)(25,40)(28,41)(32,33)(34,39)(35,44)(38,46)(42,45);(2,5)(6,8)(10,14)(11,16)(17,19,22,27,21,28,31,41)(18,24,26,39,23,34,36,37)(20,29,33,42,32,43,30,45)(25,38,40,48,35,46,44,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,23)(19,28)(25,35)(26,36)(27,41)(29,43)(40,44)(42,45);(17,20,31,30,21,32,22,33)(18,25,36,44,23,35,26,40)(19,29,41,45,28,43,27,42)(24,38,37,47,34,46,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6736104);