Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,12,11)(13,14)(15,22)(16,19)(17,21)(18,24)(20,26)(23,28)(25,27);(6,8)(9,11,12,10)(13,15)(14,18)(16,25)(17,23)(19,22)(20,28)(21,27)(24,26);(2,4)(5,6,7,8)(15,23)(18,27)(22,28)(24,25);(14,19)(15,23)(16,26)(18,28)(20,21)(22,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264202, #72264233, #72264335, … |
| maximal normal subgroup of | #72264202, #72264233, #72264335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275311);