#7090649

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,29,33,46)(20,25,31,45)(21,26,34,47)(22,37,30,28)(24,39,42,32)(35,41,48,44)(36,40,38,43);(9,10)(11,15)(12,14)(13,16)(17,19,30,39,23,33,22,32)(18,24,37,46,27,42,28,29)(20,34,48,36,31,21,35,38)(25,43,41,26,45,40,44,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,31)(18,25,27,45)(19,21,33,34)(22,35,30,48)(24,40,42,43)(26,46,47,29)(28,44,37,41)(32,36,39,38);(11,16)(13,15)(17,21,23,34)(18,26,27,47)(19,31,33,20)(22,38,30,36)(24,41,42,44)(25,29,45,46)(28,40,37,43)(32,35,39,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(20,31)(21,34)(24,42)(28,37)(29,46)(35,48)(36,38)

Polycyclic presentation:

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

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, 6997845);