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 | |
4157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,4,8)(5,13,14,9)(11,15,12,16)(18,20)(23,26)(24,28)(30,32)(33,34,38,42)(35,40,36,39)(37,41,45,48)(43,47,44,46);(1,2,7,6)(3,10,4,8)(5,9,14,13)(11,16,12,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3,7,4)(2,8,6,10)(5,11,14,12)(9,15,13,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,38,36)(34,39,42,40)(37,44,45,43)(41,47,48,46);(1,4,7,3)(2,8,6,10)(5,12,14,11)(9,15,13,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,14)(2,9,6,13)(3,11,4,12)(8,15,10,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,6,7,2)(3,8,4,10)(5,13,14,9)(11,15,12,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8092013);