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 | |
4128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18,24,26)(19,22,23,32)(20,28,21,25)(27,31,29,30)(33,34,38,41)(35,39,36,43)(37,40,46,45)(42,47,44,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,24,23)(18,22,26,32)(20,29,21,27)(25,31,28,30)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20,24,21)(18,25,26,28)(19,27,23,29)(22,30,32,31)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,21,24,20)(18,25,26,28)(19,27,23,29)(22,31,32,30)(33,34)(35,39)(36,43)(37,45)(38,41)(40,46)(42,48)(44,47);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,22)(18,19)(20,30)(21,31)(23,26)(24,32)(25,27)(28,29)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,23,24,19)(18,22,26,32)(20,29,21,27)(25,30,28,31)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10353634);