#6832778

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fh,d^2=i,e^2=ghi,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, 6739974);