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 | |
4111 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)(3,9)(4,10)(5,13)(6,8)(7,12)(11,15)(14,16)(17,18,22,24)(19,23,25,20)(21,26,30,31)(27,29,32,28)(35,41)(40,47)(42,45)(43,48);(1,3)(2,7)(4,11)(5,10)(6,12)(8,9)(13,14)(15,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,29,31,28)(33,34,38,39)(35,40,41,47)(36,44,46,37)(42,43,45,48);(2,8)(5,15)(7,9)(10,16)(18,24)(19,25)(26,31)(27,32)(34,39)(37,44)(40,47)(43,48);(1,3)(2,9)(4,13)(5,10)(6,12)(7,8)(11,14)(15,16)(17,20,22,23)(18,25,24,19)(21,29,30,28)(26,27,31,32)(33,35)(34,40)(36,45)(37,43)(38,41)(39,47)(42,46)(44,48);(1,4,6,14)(2,10,8,16)(3,11,12,13)(5,9,15,7)(17,21,22,30)(18,26,24,31)(19,27,25,32)(20,28,23,29)(33,36)(34,37)(35,42)(38,46)(39,44)(40,48)(41,45)(43,47);(1,5,6,15)(2,11,8,13)(3,10,12,16)(4,9,14,7)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,37)(34,36)(35,43)(38,44)(39,46)(40,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833590);