#9206611

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

4101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113807);