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 | |
3946 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,46)(44,48);(1,3,7,14)(2,8,12,13)(4,5,11,15)(6,10,9,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(34,39)(36,40)(41,45)(44,46);(18,20)(19,26)(25,28)(27,29)(33,35,38,43)(34,40,39,36)(37,42,47,48)(41,46,45,44);(1,4,7,11)(2,9,12,6)(3,5,14,15)(8,16,13,10)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,5)(2,10)(3,11)(4,14)(6,13)(7,15)(8,9)(12,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,36)(34,35)(37,46)(38,40)(39,43)(41,48)(42,45)(44,47);(1,6,7,9)(2,11,12,4)(3,10,14,16)(5,8,15,13)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633701);