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 | |
3945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,13,7)(4,5,9,15)(11,14,16,12)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,39,45)(35,44,38,40)(36,42,47,41)(37,43,48,46);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,16)(14,15)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,35,39,38)(34,40,45,44)(36,43,47,46)(37,41,48,42);(18,20)(19,26)(25,28)(27,29)(33,36,39,47)(34,41,45,42)(35,43,38,46)(37,40,48,44);(1,4,6,9)(2,5,10,15)(3,11,13,16)(7,12,8,14)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,36)(34,42)(35,43)(37,40)(38,46)(39,47)(41,45)(44,48);(1,3)(2,8)(4,11)(5,14)(6,13)(7,10)(9,16)(12,15)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,37)(34,43)(35,42)(36,40)(38,41)(39,48)(44,47)(45,46);(1,5)(2,9)(3,12)(4,10)(6,15)(7,16)(8,11)(13,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,38,39,35)(34,44,45,40)(36,46,47,43)(37,42,48,41)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=i,f^2=hi,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, 9633642);