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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,24,28)(19,25,22,23)(20,27,21,26)(29,32,30,31)(34,39)(36,45)(40,46)(43,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19)(18,25)(20,30)(21,29)(22,24)(23,28)(26,32)(27,31)(35,42)(40,46)(43,48)(44,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,24,21)(18,26,28,27)(19,29,22,30)(23,32,25,31)(33,34,38,39)(35,40,42,46)(36,41,45,37)(43,47,48,44);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,21,24,20)(18,27,28,26)(19,29,22,30)(23,32,25,31)(33,35)(34,40)(36,43)(37,44)(38,42)(39,46)(41,47)(45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,22,24,19)(18,25,28,23)(20,30,21,29)(26,31,27,32)(33,36)(34,41)(35,43)(37,39)(38,45)(40,47)(42,48)(44,46);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,23,24,25)(18,19,28,22)(20,31,21,32)(26,30,27,29)(33,37,38,41)(34,36,39,45)(35,44,42,47)(40,43,46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324279);