Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,18,22)(15,19,17,20)(16,21,25,28)(23,26,24,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,17)(14,19,22,20)(16,24,25,23)(21,27,28,26);(2,4)(14,20)(15,17)(16,25)(19,22)(21,26)(27,28);(1,2)(3,4)(13,16,18,25)(14,21,22,28)(15,23,17,24)(19,26,20,27);(2,4)(16,25)(21,28)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481895);