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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,26)(20,29)(21,24)(22,30)(23,25)(27,28)(31,32)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,21)(18,24,25,26)(20,27,30,31)(22,28,29,32)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,6)(7,9)(8,12)(14,15)(18,25)(20,30)(24,26)(27,31)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,22)(19,27)(21,31)(23,30)(24,28)(25,29)(26,32)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,19)(18,24,25,26)(20,31,30,27)(22,28,29,32)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,30)(26,31)(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=hi,c^2=gh,d^2=gh,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, 9701487);