Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3198 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)(13,14)(15,22)(16,25)(17,20)(18,27)(19,21)(23,24)(26,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,26)(14,16,21,23)(17,25,18,24)(20,28,27,22);(6,9)(7,11)(13,16)(14,15)(17,22)(18,28)(19,23)(20,24)(21,26)(25,27);(5,7)(6,10)(8,11)(9,12)(13,14)(15,23)(16,26)(17,27)(18,20)(19,21)(22,24)(25,28);(6,9)(7,11)(13,17,19,18)(14,20,21,27)(15,24,26,25)(16,22,23,28)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6503101);