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 | |
4283 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,27)(20,23)(21,25)(22,31)(24,29)(26,32)(28,30)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,6)(7,14)(8,11)(15,16)(18,20)(24,27)(25,30)(26,31)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,18)(19,27,29,24)(21,30,28,25)(22,31,32,26)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=hi,e^2=gh,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, 9743076);