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 | |
4124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,30)(26,32)(29,31)(33,34)(35,41)(36,43)(37,45)(38,40)(39,47)(42,44)(46,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19,21,27)(18,22,25,23)(20,29,30,26)(24,32,28,31)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(2,6)(7,10)(8,12)(14,16)(17,18,21,25)(19,23,27,22)(20,24,30,28)(26,32,29,31)(33,34,38,40)(35,36,43,41)(37,39,47,45)(42,44,46,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(33,36,38,41)(34,35,40,43)(37,44,47,48)(39,42,45,46);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,32)(28,31)(29,30)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,48)(46,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,28)(27,29)(33,37,38,47)(34,39,40,45)(35,42,43,46)(36,44,41,48)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8762629);