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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,14)(4,5,12,13)(9,10,15,16)(17,18)(19,22)(20,21)(23,24)(25,31)(26,27)(28,30)(29,32)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,47,46,43);(1,3,6,11)(2,7,8,14)(4,9,12,15)(5,10,13,16)(17,19,23,25)(18,22,24,31)(20,26,28,29)(21,27,30,32)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,8)(4,12)(7,14)(9,15)(18,24)(19,25)(20,28)(27,32)(33,36,40,38)(34,42,44,37)(35,43,48,47)(39,45,46,41);(1,4)(2,5)(3,9)(6,12)(7,10)(8,13)(11,15)(14,16)(17,20)(18,21)(19,26)(22,27)(23,28)(24,30)(25,29)(31,32)(33,37,40,42)(34,36,44,38)(35,46,48,39)(41,43,45,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,21)(18,20)(19,27)(22,26)(23,30)(24,28)(25,32)(29,31)(33,38,40,36)(34,42,44,37)(35,47,48,43)(39,45,46,41);(1,5)(2,4)(3,10)(6,13)(7,9)(8,12)(11,16)(14,15)(17,22)(18,25)(19,24)(20,27)(21,29)(23,31)(26,30)(28,32)(33,39,40,46)(34,43,44,47)(35,42,48,37)(36,45,38,41)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=g,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, 9689996);