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 |
| prime radical | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,10,5,9)(4,13,14,6)(11,12,15,16)(17,18,23,26)(19,21,24,20)(22,25,31,30)(27,29,32,28)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,8)(4,14)(9,10)(11,15)(17,19,23,24)(18,20,26,21)(22,27,31,32)(25,28,30,29)(35,42)(39,46)(43,47)(44,48);(1,3,7,5)(2,9,8,10)(4,11,14,15)(6,12,13,16)(17,20,23,21)(18,19,26,24)(22,28,31,29)(25,27,30,32)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,4,7,14)(2,6,8,13)(3,11,5,15)(9,12,10,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,7,3)(2,10,8,9)(4,15,14,11)(6,16,13,12)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27);(1,6)(2,4)(3,12)(5,16)(7,13)(8,14)(9,11)(10,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332599);