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 | |
4086 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,27,28,23)(20,25,30,32)(21,31,29,26)(33,34)(35,41)(36,37)(38,42)(39,48)(40,44)(43,45)(46,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35)(34,41)(36,43)(37,45)(38,48)(39,42)(40,46)(44,47);(2,7)(3,10)(8,15)(11,12)(18,24)(23,27)(25,32)(26,31)(33,36,40,38)(34,42,44,37)(35,43,46,48)(39,47,45,41);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,37,40,42)(34,36,44,38)(35,45,46,39)(41,43,47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,38,40,36)(34,42,44,37)(35,43,46,48)(39,41,45,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,39,40,45)(34,43,44,48)(35,42,46,37)(36,47,38,41)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9613502);