#6820826

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

1320 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)(20,32)(21,23)(22,26)(24,31)(25,35)(28,38)(29,37)(30,39)(33,41)(34,42)(36,45)(40,44)(43,48)(46,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,33)(22,29)(24,38)(25,41)(27,40)(31,44)(32,45)(35,36)(39,42)(43,46);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,29,44,21,31,22,28)(18,24,37,38,23,27,26,40)(20,34,33,43,35,30,36,46)(25,42,45,48,32,39,41,47);(10,14)(13,15)(18,23)(20,35)(24,27)(26,37)(30,34)(33,36)(38,40)(43,46);(17,20,29,33,21,35,22,36)(18,25,37,45,23,32,26,41)(19,30,44,46,31,34,28,43)(24,39,38,47,27,42,40,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gh,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,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, 6728022);