Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1810 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)(6,7)(10,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,27)(23,28)(24,26);(5,6)(7,8)(13,15)(14,18)(16,22)(17,24)(19,27)(20,26)(21,28)(23,25);(2,4)(5,7,8,6)(14,19)(15,22)(16,25)(20,21)(23,24)(27,28);(9,10,11,12)(14,20)(18,26)(19,21)(27,28)Polycyclic presentation:
2^9|c^2=e,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264459, #72264485, #72264591, … |
| maximal normal subgroup of | #72264459, #72264485, #72264591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319245);