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 | |
4010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,11)(5,8,14,16)(10,12,13,15)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32)(33,34,39,43)(35,38,45,40)(36,46,44,41)(37,47,48,42);(1,3)(2,4)(5,13)(6,7)(8,15)(9,11)(10,14)(12,16)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32)(33,35,39,45)(34,40,43,38)(36,37,44,48)(41,47,46,42);(3,7)(4,11)(10,13)(12,15)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,36,39,44)(34,41,43,46)(35,37,45,48)(38,42,40,47);(1,4,6,11)(2,7,9,3)(5,12,14,15)(8,13,16,10)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,37,39,48)(34,42,43,47)(35,44,45,36)(38,46,40,41);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,35)(34,40)(36,37)(38,43)(39,45)(41,47)(42,46)(44,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,16)(11,15)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,38,39,40)(34,35,43,45)(36,42,44,47)(37,46,48,41)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=gh,f^2=h,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, 8748554);