#7094746

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

1251 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,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=dfi,e^b=ehi,f^b=fi,d^c=dhi,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, 7001942);