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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2063 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)(15,20)(16,22)(17,23)(18,27)(19,24)(21,25)(26,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,18)(17,26)(19,25)(21,24)(22,27)(23,28);(2,4)(13,16,19,26)(14,21,24,20)(15,17,25,18)(22,23,28,27);(1,2)(3,4)(13,16)(14,22)(15,17)(18,25)(19,26)(20,23)(21,27)(24,28);(2,4)(13,17,19,18)(14,23,24,27)(15,16,25,26)(20,22,21,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^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, 6481915);