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 | #45824 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,48)(46,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,35,39,48)(34,40,44,45)(36,46,43,42)(37,38,47,41);(2,6)(3,9)(4,12)(14,16)(18,23)(20,26)(21,30)(27,31)(36,43)(37,47)(38,41)(42,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,26)(18,24,23,19)(21,28,30,32)(25,31,29,27)(33,36,39,43)(34,41,44,38)(35,46,48,42)(37,45,47,40);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,37,39,47)(34,42,44,46)(35,41,48,38)(36,40,43,45);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)(33,38,39,41)(34,43,44,36)(35,47,48,37)(40,42,45,46)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964627);