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 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,13,8)(4,14,11,6)(5,10,15,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,9)(3,13)(4,11)(10,16)(18,20)(23,25)(24,28)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=g,e^2=gi,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, 9746427);