#7091910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,29,32,46)(20,33,30,25)(21,26,34,47)(22,38,40,28)(24,41,44,31)(35,43,36,48)(37,45,39,42);(9,10)(11,15)(12,14)(13,16)(17,19,22,31,23,32,40,41)(18,24,28,29,27,44,38,46)(20,34,35,39,30,21,36,37)(25,45,43,26,33,42,48,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,30)(18,25,27,33)(19,21,32,34)(22,35,40,36)(24,42,44,45)(26,29,47,46)(28,43,38,48)(31,37,41,39);(11,16)(13,15)(17,21,23,34)(18,26,27,47)(19,30,32,20)(22,39,40,37)(24,43,44,48)(25,46,33,29)(28,45,38,42)(31,35,41,36);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(20,30)(21,34)(24,44)(28,38)(29,46)(35,36)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6999106);