#7093538

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	#30512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,31)(20,26)(21,27)(22,29)(23,28)(24,30)(25,33)(32,42)(34,45)(35,46)(36,44)(37,40)(38,43)(39,48)(41,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33,24,21,22,37)(18,25,29,31,30,40,28,27)(20,34,35,47,36,32,38,48)(26,41,43,45,44,39,46,42);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,36)(18,26,30,44)(19,32,21,34)(22,38,23,35)(25,39,40,41)(27,45,31,42)(28,46,29,43)(33,48,37,47);(11,16)(13,15)(17,21,24,19)(18,27,30,31)(20,32,36,34)(22,37,23,33)(25,29,40,28)(26,42,44,45)(35,47,38,48)(39,46,41,43);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,21,37)(20,35,36,38)(25,31,40,27)(26,43,44,46)(32,47,34,48)(39,45,41,42)

Polycyclic presentation:

2^9|b^2=fg,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, 7000734);