Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,19,17,22,16,21)(15,20,25,26,23,27,24,28);(1,2)(3,4)(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(16,18)(20,26)(21,22)(24,25)(27,28);(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,28)(22,27);(15,23)(20,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=gi,c^2=f,g^2=i,b^a=bf,c^a=cg,c^b=ch,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339403);