#6820799

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

1295 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)(20,31)(21,23)(22,26)(24,29)(25,34)(28,37)(30,38)(32,40)(33,36)(35,45)(39,42)(41,44)(43,48)(46,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,32)(22,33)(24,37)(25,40)(27,39)(29,42)(31,45)(34,35)(38,41)(43,46);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,22,28,21,29,33,42)(18,24,26,39,23,27,36,37)(20,30,35,43,34,44,32,46)(25,38,40,48,31,41,45,47);(10,14)(13,15)(18,23)(19,29)(25,31)(26,36)(28,42)(30,44)(40,45)(43,46);(17,20,33,32,21,34,22,35)(18,25,36,45,23,31,26,40)(19,30,42,46,29,44,28,43)(24,38,37,47,27,41,39,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727995);