#7091212

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

1258 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)(17,18,22,25)(19,27,30,39)(20,32,34,24)(21,36,37,26)(23,29,40,31)(28,44,33,43)(35,42,48,47)(38,45,41,46);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,25,40)(20,33,34,28)(21,29,37,31)(24,41,32,38)(26,39,36,27)(35,45,48,46)(42,43,47,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,22,34)(18,24,25,32)(19,28,30,33)(21,35,37,48)(23,38,40,41)(26,42,36,47)(27,43,39,44)(29,46,31,45);(11,16)(13,15)(19,29)(20,34)(21,37)(23,39)(24,32)(26,36)(27,40)(28,45)(30,31)(33,46)(38,43)(41,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,25)(19,30)(20,34)(26,36)(29,31)(35,48)(38,41)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,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=eh,f^b=fi,d^c=dh,e^c=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, 6998408);