Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,22,26,20,24)(19,27,28,32,30,31,29,23);(1,2,3,4)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,31)(26,32);(10,13)(14,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,27,32,31);(6,8)(9,11,12,15)(10,14,13,16)(18,25)(20,21)(23,31)(24,26)(27,32)(28,29)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^d=ei,g^d=gi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264578, #72264588, #72264598, … |
| maximal normal subgroup of | #72264578, #72264588, #72264598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300744);