#7092061

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(17,18,23,28)(19,29,34,47)(20,25,21,26)(22,38,30,27)(24,39,42,33)(31,46,36,41)(32,44,48,40)(35,45,37,43);(9,10)(11,15)(12,14)(13,16)(17,19,30,39,23,34,22,33)(18,24,38,47,28,42,27,29)(20,31,37,32,21,36,35,48)(25,40,45,41,26,44,43,46);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,25,28,26)(19,31,34,36)(22,35,30,37)(24,40,42,44)(27,43,38,45)(29,46,47,41)(32,33,48,39);(11,16)(13,15)(17,21,23,20)(18,26,28,25)(19,32,34,48)(22,35,30,37)(24,41,42,46)(27,43,38,45)(29,44,47,40)(31,33,36,39);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,21)(25,26)(31,36)(32,48)(35,37)(40,44)(41,46)(43,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=ghi,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=eh,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, 6999257);