Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,9)(7,11,12,10)(13,14)(15,20)(16,18)(17,26)(19,22)(21,24)(23,28)(25,27);(1,2,4,3)(6,9)(10,11)(13,15,18,17)(14,20,16,26)(19,25,28,24)(21,22,27,23);(2,3)(5,7,8,12)(6,10,9,11)(13,16,18,14)(15,21,17,27)(19,22,28,23)(20,25,26,24);(1,3,4,2)(7,12)(10,11)(13,17,18,15)(14,21,16,27)(19,25,28,24)(20,22,26,23)Polycyclic presentation:
2^9|a^2=fi,b^2=ef,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367270);