#7090474

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	#23857	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,28)(21,24)(25,38)(26,31)(27,36)(29,43)(32,34)(35,40)(39,47)(41,42)(44,48);(9,10)(11,15)(12,14)(13,16)(17,18,24,36,22,27,21,23)(19,25,37,38,28,40,30,35)(20,31,34,42,33,41,32,26)(29,44,46,48,43,47,45,39);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,25,27,40)(20,29,33,43)(21,30,24,37)(23,35,36,38)(26,39,42,48)(31,44,41,47)(32,45,34,46);(11,16)(13,15)(17,20,22,33)(18,26,27,42)(19,29,28,43)(21,34,24,32)(23,31,36,41)(25,39,40,48)(30,46,37,45)(35,44,38,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,28)(25,40)(29,43)(30,37)(35,38)(39,48)(44,47)(45,46)

Polycyclic presentation:

2^9|b^2=fi,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=eh,d^c=dh,e^c=ei,f^d=fi

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, 6997670);