Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,24,23,28,21,27)(19,25,20,26,30,31,29,32);(2,3)(9,11,12,16)(10,13,14,15)(18,24)(19,20)(21,22)(25,31)(27,28)(29,30);(10,14)(13,15)(17,19,21,29,23,30,22,20)(18,25,27,32,28,31,24,26);(1,3,4,2)(6,8)(11,16)(13,15)(17,20,21,19,23,29,22,30)(18,26,27,25,28,32,24,31)Polycyclic presentation:
2^9|a^2=ei,b^2=f,c^2=gi,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351537);