Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| 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 | #14407 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,22)(3,12,14,8,16,24,15,25)(4,17,20,31,21,9,19,27)(13,26,18,23,29,30,28,32);(1,3)(2,8)(4,18)(5,15)(6,16)(7,14)(9,26)(10,24)(11,25)(12,22)(13,19)(17,30)(20,29)(21,28)(23,27)(31,32);(1,2,6,11)(3,8)(4,9,21,17)(5,22,7,10)(12,15)(13,23)(14,24)(16,25)(18,30)(19,31,20,27)(26,28)(29,32);(1,4,6,21)(2,9,11,17)(3,13)(5,19,7,20)(8,23)(10,27,22,31)(12,26)(14,18)(15,28)(16,29)(24,30)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ef,e^2=i,f^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,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, 292730);