Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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 | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(15,24)(16,17)(19,27)(20,28)(21,22)(25,26);(6,9)(10,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,25)(24,28)(26,27);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,27)(22,28);(13,15)(14,20)(16,24)(17,26)(18,25)(19,23)(21,28)(22,27);(13,16,18,17)(14,21,23,22)(15,24,25,26)(19,27,20,28)Polycyclic presentation:
2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=ci,g^c=gi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536421);