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 | |
4238 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)(3,8)(4,9)(5,10)(6,7)(11,12)(13,15)(14,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,46)(43,47)(44,48);(2,7)(3,10)(9,15)(11,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(34,39)(40,42)(41,45)(47,48);(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,35,38,36)(34,40,39,42)(37,43,46,44)(41,47,45,48);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36,38,35)(34,40,39,42)(37,43,46,44)(41,48,45,47);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48)Polycyclic presentation:
2^9|c^2=h,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10426553);