#6820814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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)(24,30)(28,38)(29,42)(31,45)(32,39)(33,36)(34,37)(35,41)(40,44)(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)(24,38)(25,41)(27,40)(30,44)(33,35)(36,45)(39,42)(43,46);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,22,28,21,30,34,44)(18,24,26,40,23,27,37,38)(20,32,35,46,33,29,31,43)(25,42,41,47,36,39,45,48);(10,14)(13,15)(18,23)(20,33)(24,27)(26,37)(29,32)(31,35)(38,40)(43,46);(17,20,34,31,21,33,22,35)(18,25,37,45,23,36,26,41)(19,29,44,46,30,32,28,43)(24,39,38,47,27,42,40,48)

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,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6728010);