Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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×#3482 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3108 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)(13,14,16,22,18,21,17,23)(15,24,26,20,27,28,25,19);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,21)(22,28)(23,24);(6,9)(7,11)(13,15)(14,20)(16,25)(17,26)(18,27)(19,21)(22,24)(23,28);(5,7)(6,10)(8,11)(9,12)(14,21)(19,20)(22,23)(24,28);(6,9)(7,11)(13,16,18,17)(14,22,21,23)(15,25,27,26)(19,28,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6503609);