#7072868

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

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

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, 6980064);