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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,8)(7,10,12,11)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,27)(26,28);(1,2,3,4)(6,8)(10,11)(13,15)(14,18)(16,23)(17,24)(19,26)(20,22)(21,27)(25,28);(5,6,9,8)(7,11,12,10)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(5,7,9,12)(6,10,8,11)(15,23)(18,22)(24,28)(26,27);(5,8,9,6)(7,10,12,11)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=h,d^2=h,e^2=h,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, 6534579);