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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,27)(17,22)(18,23)(19,24)(20,28)(21,26);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,21,24,25)(16,18,17,20)(22,23,27,28);(2,4)(13,16)(14,21)(15,20)(17,19)(18,26)(22,23)(24,25)(27,28);(1,2)(3,4)(13,17,19,16)(14,22,24,27)(15,18,26,20)(21,28,25,23);(2,4)(13,18,19,20)(14,23,24,28)(15,16,26,17)(21,22,25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6481929);