Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2568 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)(10,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(2,4)(13,15)(14,18)(16,21)(17,23)(19,26)(20,25)(22,24)(27,28);(6,8)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(9,10,11,12)(14,19)(18,26)(20,28)(25,27)Polycyclic presentation:
2^9|d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264632, #72264653, #72264844, … |
| maximal normal subgroup of | #72264632, #72264653, #72264844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318768);