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 | |
4274 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)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34)(35,43)(36,41)(37,46)(38,40)(39,42)(44,48)(45,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35,39,38)(34,40,42,43)(36,44,37,45)(41,47,46,48);(2,7)(6,13)(8,11)(14,15)(18,25)(19,30)(20,27)(28,32)(33,36,39,37)(34,41,42,46)(35,44,38,45)(40,47,43,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,37,39,36)(34,41,42,46)(35,45,38,44)(40,47,43,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,34)(35,40)(36,41)(37,46)(38,43)(39,42)(44,47)(45,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,38,39,35)(34,40,42,43)(36,45,37,44)(41,47,46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=i,f^2=g,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, 10065673);