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 | |
4044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,34,40,43)(35,44,37,41)(36,39,38,42)(45,48,46,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,31,29,28)(33,35,40,37)(34,41,43,44)(36,45,38,46)(39,47,42,48);(2,6)(3,9)(8,11)(10,12)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31)(33,36,40,38)(34,42,43,39)(35,45,37,46)(41,48,44,47);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,32)(29,30)(33,37,40,35)(34,41,43,44)(36,45,38,46)(39,48,42,47);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,29,32,27)(33,38,40,36)(34,42,43,39)(35,46,37,45)(41,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21,22,30)(18,25,26,32)(19,27,24,29)(20,28,23,31)(33,39,40,42)(34,36,43,38)(35,47,37,48)(41,45,44,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838232);