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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4082 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,16)(12,15)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31)(33,34,38,39)(35,43,45,40)(36,44,46,41)(37,47,48,42);(2,7)(4,9)(8,10)(11,15)(19,26)(20,25)(21,30)(23,29)(34,39)(36,46)(40,43)(42,47);(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32)(33,35,38,45)(34,40,39,43)(36,42,46,47)(37,44,48,41);(1,3,6,13)(2,8,7,10)(4,11,9,15)(5,12,14,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36,38,46)(34,41,39,44)(35,42,45,47)(37,43,48,40);(1,4,6,9)(2,5,7,14)(3,11,13,15)(8,12,10,16)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,46)(47,48);(1,5)(2,9)(3,12)(4,7)(6,14)(8,15)(10,11)(13,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(33,37)(34,42)(35,44)(36,43)(38,48)(39,47)(40,46)(41,45)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015916);