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 | |
4151 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)(3,8,10,5)(4,9,12,15)(11,13,16,14)(17,18)(19,23)(20,26)(21,29)(22,24)(25,28)(27,32)(30,31)(33,34)(35,43)(36,37)(38,44)(39,41)(40,46)(42,48)(45,47);(2,7)(3,10)(4,12)(13,14)(17,19,22,28)(18,23,24,25)(20,29,31,27)(21,30,32,26)(33,35,39,46)(34,40,41,43)(36,42,47,44)(37,38,45,48);(18,24)(19,28)(26,30)(27,29)(34,41)(36,47)(40,43)(42,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,13)(12,16)(14,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,32)(27,31)(33,36)(34,37)(35,44)(38,43)(39,47)(40,48)(41,45)(42,46);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,13)(10,16)(17,20)(18,26)(19,29)(21,23)(22,31)(24,30)(25,32)(27,28)(33,37)(34,36)(35,38)(39,45)(40,42)(41,47)(43,44)(46,48);(1,5,6,8)(2,3,7,10)(4,13,12,14)(9,16,15,11)(17,21,22,32)(18,27,24,29)(19,30,28,26)(20,25,31,23)(33,38)(34,42)(35,45)(36,43)(37,46)(39,48)(40,47)(41,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9844977);