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 | |
4100 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(19,25)(22,26)(27,31)(30,32)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(2,7)(4,8)(5,14)(11,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,29)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8738372);