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 | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(14,18,20,23)(15,21,22,24);(1,2,3,4)(13,14,16,23,17,20,25,18)(15,19,24,28,22,27,21,26);(6,9)(10,12)(15,22)(19,27)(21,24)(26,28);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,24)(17,22)(18,26)(20,27)(21,25)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,e^2=i,g^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300733);