Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,10)(11,12)(13,14,17,19)(15,18,21,27)(16,23,24,22)(20,25,28,26);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(14,19)(15,21)(16,24)(25,26);(2,4)(5,7,9,8)(6,11,10,12)(13,15,18,19,17,21,27,14)(16,22,25,28,24,23,26,20);(5,8,9,7)(6,11,10,12)(13,16,17,24)(14,20,19,28)(15,22,21,23)(18,25,27,26)Polycyclic presentation:
2^9|a^2=i,c^2=efi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331335);