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 | |
4139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(19,25)(22,29)(26,31)(30,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,7)(3,9)(8,15)(10,16)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5)(2,3)(4,11)(6,14)(7,9)(8,16)(10,15)(12,13)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9758569);