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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,8)(9,10,11,12)(13,14)(15,18)(16,19)(17,21)(20,24)(22,25)(23,26)(27,28);(1,2,3,4)(13,15)(14,18)(16,23)(17,22)(19,26)(20,27)(21,25)(24,28);(5,6,7,8)(15,22)(18,25)(23,28)(26,27);(14,19)(16,24)(18,26)(20,21)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264291, #72264315, #72264386, … |
| maximal normal subgroup of | #72264291, #72264315, #72264386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299645);