#9232783

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9139979);