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 | |
4097 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,34,39,43)(35,40,45,38)(36,41,44,48)(37,42,46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,35)(34,40)(36,46)(37,44)(38,43)(39,45)(41,47)(42,48);(2,3)(7,9)(8,10)(14,15)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,36,39,44)(34,41,43,48)(35,37,45,46)(38,47,40,42);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,46)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,37)(34,42)(35,44)(36,45)(38,41)(39,46)(40,48)(43,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,38,39,40)(34,35,43,45)(36,47,44,42)(37,48,46,41)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10102776);