#7171986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7079182);