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 | |
4252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,18,21,24)(19,22,25,26)(20,23,28,30)(27,29,31,32)(33,34,39,42)(35,40,44,38)(36,45,37,41)(43,48,46,47);(2,4)(7,9)(8,12)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35)(34,40)(36,46)(37,43)(38,42)(39,44)(41,48)(45,47);(2,4)(5,13)(7,9)(10,16)(19,25)(20,28)(22,26)(23,30)(33,36,39,37)(34,41,42,45)(35,43,44,46)(38,48,40,47);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,37,39,36)(34,41,42,45)(35,43,44,46)(38,47,40,48);(1,4,6,2)(3,7,11,9)(5,12,13,8)(10,14,16,15)(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32)(33,38)(34,35)(36,47)(37,48)(39,40)(41,46)(42,44)(43,45);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,20,21,28)(18,23,24,30)(19,27,25,31)(22,29,26,32)(33,35)(34,38)(36,46)(37,43)(39,44)(40,42)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851052);