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 | |
4007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,12,13,8)(11,14,16,15)(17,18,22,25)(19,20,26,23)(21,29,30,24)(27,31,32,28)(34,38)(36,43)(39,44)(42,47);(2,4)(7,9)(8,12)(14,15)(19,26)(20,23)(27,32)(28,31)(35,41)(39,44)(42,47)(46,48);(3,10)(7,9)(11,16)(14,15)(17,19,22,26)(18,23,25,20)(21,27,30,32)(24,31,29,28)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,30)(18,24,25,29)(19,27,26,32)(20,28,23,31)Polycyclic presentation:
2^9|a^2=gi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022544);