Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119362 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,28,31,43)(20,33,37,25)(21,26)(22,38)(23,27)(24,30,40,32)(29,41)(34,39)(35,47,36,42)(44,48)(45,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,34)(21,32)(22,30)(23,31)(25,41)(26,43)(27,40)(28,38)(29,36)(33,46)(35,45)(37,44)(39,42)(47,48);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,20,21,35,23,37,22,36)(18,25,26,42,27,33,38,47)(19,29,30,44,31,45,32,34)(24,39,28,41,40,48,43,46)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,e^c=eh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26558);