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 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,13,14,8)(4,5,11,10)(6,16,15,12)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,34,40,37)(35,44,46,41)(36,47,38,42)(39,48,45,43);(1,3)(2,8)(4,15)(5,16)(6,11)(7,14)(9,13)(10,12)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(33,35)(34,41)(36,45)(37,44)(38,39)(40,46)(42,43)(47,48);(2,9)(3,14)(4,11)(12,16)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,36,40,38)(34,42,37,47)(35,45,46,39)(41,43,44,48);(1,4,7,11)(2,10,9,5)(3,15,14,6)(8,12,13,16)(17,21,23,20)(18,26,27,25)(19,30,22,29)(24,32,28,31)(33,37,40,34)(35,44,46,41)(36,42,38,47)(39,48,45,43);(1,5,7,10)(2,11,9,4)(3,12,14,16)(6,8,15,13)(17,22,23,19)(18,24,27,28)(20,30,21,29)(25,31,26,32)(33,38,40,36)(34,42,37,47)(35,45,46,39)(41,48,44,43);(1,6,7,15)(2,12,9,16)(3,11,14,4)(5,8,10,13)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,31)(30,32)(33,39,40,45)(34,43,37,48)(35,38,46,36)(41,42,44,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=h,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, 9630350);