Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14391 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,4,7,11)(3,13,17,19,18,14,16,9)(5,20,22,27,23,10,24,28)(15,26,30,32,31,29,21,25);(1,3)(2,9)(4,19)(5,21)(6,17)(7,16)(8,18)(10,26)(11,14)(12,13)(15,22)(20,29)(23,30)(24,31)(25,27)(28,32);(1,4)(2,8)(3,14)(5,20)(6,11)(7,12)(9,17)(10,23)(13,18)(15,29)(16,19)(21,32)(22,27)(24,28)(25,30)(26,31);(1,5,8,23)(2,10,4,20)(3,15)(6,22,7,24)(9,25)(11,27,12,28)(13,26)(14,29)(16,21)(17,30)(18,31)(19,32)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292061);