#9232786

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

4157 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 9139982);