Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184773 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,18,24,26)(15,21,22,28,23,19,25,27);(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(16,24)(19,21)(22,25);(13,15,17,23)(14,19,18,21)(16,25,24,22)(20,28,26,27)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^c=hi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72302360, #72302839, #72302843 |
| maximal normal subgroup of | #72302360, #72302839, #72302843 |
| maximal quotient of | #72303084 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299446);