Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500092 |
| 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×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,7,10,12,8,11)(13,14)(15,18)(16,20)(17,19)(21,25)(22,24)(23,26)(27,28);(2,4)(5,7,10,11)(6,8,12,9)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,6)(7,8)(9,11)(10,12)(14,19)(15,21)(16,23)(24,28);(5,8,10,9)(6,7,12,11)(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,24,25,28);(5,8,10,9)(6,7,12,11)(15,21)(18,25)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431982);