#6820779

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1307 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=ghi,e^2=fhi,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, 6727975);