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 | |
4280 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,6)(7,9)(8,13)(14,15)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(34,35)(39,41)(40,42)(46,47);(4,12)(8,13)(10,16)(14,15)(18,23)(22,25)(24,28)(31,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=i,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8179153);