Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1096 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)(5,6,9,8)(7,11,12,10)(13,14)(15,23)(16,21)(17,22)(18,27)(19,25)(20,26)(24,28);(5,7,9,12)(6,10,8,11)(13,14,16,21)(15,24,25,20)(17,27,18,22)(19,28,23,26);(2,4)(13,15)(14,19)(16,25)(17,28)(18,26)(20,22)(21,23)(24,27);(5,8,9,6)(7,10,12,11)(13,15)(14,20)(16,25)(17,26)(18,28)(19,22)(21,24)(23,27)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327405);