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 |
| prime radical | #45805 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,16)(12,15)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(36,43)(40,45)(42,47)(46,48);(2,7)(4,9)(8,10)(11,15)(18,21)(19,25)(20,29)(30,32)(34,38)(35,41)(40,45)(42,47);(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,6,13)(2,8,7,10)(4,11,9,15)(5,12,14,16)(17,20,23,29)(18,24,21,28)(19,27,25,31)(22,30,26,32);(1,4,6,9)(2,5,7,14)(3,11,13,15)(8,12,10,16)(17,21,23,18)(19,26,25,22)(20,28,29,24)(27,32,31,30)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5)(2,9)(3,12)(4,7)(6,14)(8,15)(10,11)(13,16)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000456);