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 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,26)(20,28)(21,31)(22,24)(23,29)(25,30)(27,32)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(2,6)(7,9)(8,12)(14,15)(18,24)(20,30)(23,26)(27,31)(33,36,39,46)(34,41,43,37)(35,38,45,44)(40,42,48,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,23,32,26)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,26)(20,27)(21,25)(22,29)(23,24)(28,32)(30,31)(33,37)(34,36)(35,42)(38,48)(39,41)(40,44)(43,46)(45,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,27)(19,28)(20,23)(22,32)(24,31)(25,29)(26,30)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,e^2=ghi,f^2=gh,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, 9770638);