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 | |
4085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,9,5,8)(4,13,15,6)(11,12,14,16)(17,18,23,26)(19,27,20,24)(21,22,25,32)(28,31,30,29)(33,34,39,36)(35,40,46,43)(37,47,45,41)(38,42,44,48);(1,3,7,5)(2,8,10,9)(4,11,15,14)(6,12,13,16)(19,20)(24,27)(28,30)(29,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(17,19,23,20)(18,24,26,27)(21,28,25,30)(22,29,32,31)(33,34,39,36)(35,43,46,40)(37,41,45,47)(38,42,44,48);(1,4,7,15)(2,6,10,13)(3,11,5,14)(8,12,9,16)(17,20,23,19)(18,24,26,27)(21,30,25,28)(22,29,32,31)(33,36,39,34)(35,43,46,40)(37,47,45,41)(38,42,44,48);(1,5,7,3)(2,9,10,8)(4,14,15,11)(6,16,13,12)(17,21,23,25)(18,22,26,32)(19,28,20,30)(24,29,27,31)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,6)(2,4)(3,12)(5,16)(7,13)(8,11)(9,14)(10,15)(17,22)(18,25)(19,29)(20,31)(21,26)(23,32)(24,30)(27,28)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378264);