Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535696 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61255 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,23,27)(19,28,30,42)(20,33,36,25)(21,26,22,37)(24,31,39,32)(29,43,44,41)(34,48,35,40)(38,46,47,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,31,23,30,22,32)(18,24,37,42,27,39,26,28)(20,29,34,45,36,44,35,46)(25,38,48,43,33,47,40,41);(3,9)(6,12)(10,15)(13,16)(17,20,22,35,23,36,21,34)(18,25,26,40,27,33,37,48)(19,29,32,46,30,44,31,45)(24,38,28,41,39,47,42,43);(1,4)(2,7)(3,10)(5,11)(6,13)(8,14)(9,15)(12,16)(19,30)(24,39)(28,42)(29,44)(31,32)(38,47)(41,43)(45,46);(4,11)(7,14)(10,15)(13,16)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,34,36,35)(24,28,39,42)(25,40,33,48)(29,45,44,46)(38,41,47,43)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473467);