Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,25)(19,23,26,30)(20,22,28,24)(27,29,32,31);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14);(3,7)(5,8)(10,11)(14,15)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,23,25,30)(20,27,28,32)(22,29,24,31);(3,7)(5,8)(10,11)(14,15)(19,26)(23,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7109756);