Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×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×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3710 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)(5,6)(7,8)(9,10,11,12)(14,18)(16,24)(19,25)(20,26)(21,28)(23,27);(13,14,16,18,17,20,24,26)(15,21,23,28,22,19,27,25);(2,4)(13,15)(14,19)(16,23)(17,22)(18,25)(20,21)(24,27)(26,28);(6,8)(15,22)(19,21)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72307672, #72307871, #72307874, … |
| maximal normal subgroup of | #72307672, #72307871, #72307874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299377);