Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(10,11)(13,14,16,18)(15,19,22,25)(17,20,24,21)(23,27,28,26);(1,2,3,4)(9,10,12,11)(14,18)(15,22)(20,21)(23,28);(2,4)(5,6,8,7)(9,10)(11,12)(13,15)(14,19)(16,22)(17,23)(18,25)(20,27)(21,26)(24,28);(5,7,8,6)(9,11,12,10)(14,20)(15,22)(17,24)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=efi,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373919);