Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384163 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14373 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,19,24)(3,11,14,25,13,7,18,21)(4,8,17,23,16,22,29,31)(12,20,27,30,15,26,28,32);(1,3)(2,7)(4,15)(5,18)(6,13)(8,20)(9,11)(10,25)(12,16)(14,19)(17,27)(21,24)(22,26)(23,32)(28,29)(30,31);(1,4)(2,8)(3,12,13,15)(5,17)(6,16)(7,20,11,26)(9,22)(10,23)(14,27,18,28)(19,29)(21,30,25,32)(24,31);(2,9)(5,19)(8,22)(14,18)(17,29)(21,25)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291360);