#7171967

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1249 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=i,d^2=f,e^2=h,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, 7079163);