Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14,17,20)(15,22,18,19)(16,24)(21,25)(23,28)(26,27);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,17,18)(14,19,20,22)(16,25,26,23)(21,24,28,27);(2,4)(5,6,9,8)(7,10,12,11)(14,20)(15,18)(21,28)(23,25);(5,8,9,6)(7,10,12,11)(13,16,17,26)(14,21,20,28)(15,23,18,25)(19,27,22,24)Polycyclic presentation:
2^9|a^2=efi,b^2=e,c^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327893);