Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31);(2,7)(3,11)(4,14)(13,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,34)(35,40)(36,37)(38,39)(41,45)(42,46)(43,48)(44,47);(2,7)(4,14)(8,10)(12,16)(21,30)(24,32)(26,29)(28,31)(34,39)(36,45)(40,47)(42,48);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,35)(34,40)(36,42)(37,46)(38,44)(39,47)(41,43)(45,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21,22,30)(18,24,25,32)(19,26,27,29)(20,28,23,31)(33,36)(34,41)(35,42)(37,39)(38,45)(40,43)(44,48)(46,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,30)(31,32)(33,37,38,41)(34,36,39,45)(35,43,44,46)(40,48,47,42)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540991);