Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,10)(7,8)(13,14,18,20,19,22,17,15)(16,24,27,21,26,23,25,28);(5,6,9,10)(7,11,8,12)(13,15,17,22,19,20,18,14)(16,23,27,28,26,24,25,21);(2,4)(5,7)(6,11)(8,9)(10,12)(14,20)(15,22)(17,18)(21,24)(23,28)(25,27);(5,8,9,7)(6,12,10,11)(13,16)(14,21)(15,23)(17,27)(18,25)(19,26)(20,24)(22,28)Polycyclic presentation:
2^9|a^2=efi,b^2=egi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372022);