Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,11)(7,10,8,12)(13,14,20,24,19,15,18,23)(16,25,26,27,28,21,22,17);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,19,14)(16,27,28,17)(18,23,20,24)(21,26,25,22);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,22,19,28,20,26)(14,21,23,27,15,25,24,17);(13,17,18,21,19,27,20,25)(14,22,23,28,15,26,24,16)Polycyclic presentation:
2^9|a^2=ef,b^2=efg,c^2=efi,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346760);