#7019124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926320);