Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| 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 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,27)(17,21)(18,23)(20,25)(22,28)(24,26);(6,9)(10,12)(13,15,18,16)(14,19,23,27)(17,26,28,25)(20,21,24,22);(1,2,3,4)(13,14)(15,24)(16,20)(17,22)(18,23)(19,26)(21,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,20,23,24)(17,26,28,25)(19,21,27,22)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=f,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305441);