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 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,9)(3,4)(10,16)(11,12)(18,20)(23,25)(24,28)(31,32)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,39,40,47)(34,42,37,45)(35,43,41,48)(36,44,38,46)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9701481);