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 | |
4058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,9,4,8)(5,10,13,14)(11,16,12,15)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10060276);