#6820778

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,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, 6727974);