Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,28)(19,30,32,24)(20,26,23,27)(22,37,36,40)(25,44,46,33)(29,35,47,39)(31,42,34,41)(38,45,48,43);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,34)(21,32)(22,33)(23,31)(25,40)(26,42)(27,41)(28,30)(29,43)(35,48)(36,44)(37,46)(38,39)(45,47);(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,40)(26,45)(27,43)(28,46)(29,41)(30,37)(33,39)(35,44)(36,48)(42,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,21,23)(18,26,28,27)(19,31,32,34)(22,35,36,39)(24,41,30,42)(25,45,46,43)(29,37,47,40)(33,38,44,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,23)(18,27,28,26)(19,31,32,34)(22,35,36,39)(24,42,30,41)(25,43,46,45)(29,40,47,37)(33,38,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72318797, #72320055, #72320093, … |
| maximal normal subgroup of | #72318797, #72320055, #72320093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531603);