#9214261

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

4217 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 9121457);