#7078842

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1387 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,35,46)(20,26,21,27)(22,28,23,39)(25,33,42,34)(31,47,37,41)(32,45,48,40)(36,43,38,44);(9,10)(11,15)(12,14)(13,16)(17,19,24,35)(18,25,29,42)(20,31,21,37)(22,34,23,33)(26,40,27,45)(28,46,39,30)(32,38,48,36)(41,43,47,44);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,38)(23,36)(25,40)(27,29)(28,44)(30,47)(32,33)(34,48)(35,37)(39,43)(41,46)(42,45);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,24,20)(18,27,29,26)(19,32,35,48)(22,38,23,36)(25,41,42,47)(28,44,39,43)(30,45,46,40)(31,34,37,33);(10,14)(13,15)(17,22,24,23)(18,28,29,39)(19,33,35,34)(20,36,21,38)(25,30,42,46)(26,43,27,44)(31,48,37,32)(40,41,45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gi,e^2=i,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, 6986038);