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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,12,13,8)(11,14,16,15)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,31)(29,30)(33,34,38,40)(35,41,43,37)(36,39,46,44)(42,47,48,45);(2,4)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,25,32)(20,28,30,27)(21,26,31,29)(33,35)(34,37)(36,42)(38,43)(39,45)(40,41)(44,47)(46,48);(2,4)(3,10)(5,13)(14,15)(18,25)(19,22)(20,30)(21,31)(33,34,38,40)(35,37,43,41)(36,44,46,39)(42,47,48,45);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,20)(18,26)(19,28)(21,32)(22,27)(23,30)(24,31)(25,29)(33,34)(35,37)(36,44)(38,40)(39,46)(41,43)(42,47)(45,48);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,21,23,31)(18,27,25,28)(19,29,22,26)(20,24,30,32)(33,36,38,46)(34,39,40,44)(35,42,43,48)(37,45,41,47);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,24,25,32)(20,27,30,28)(21,26,31,29)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8923713);