Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25733 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,17)(15,20,22,18)(16,23,27,24)(21,28,25,26);(2,3)(5,7,8,12)(6,9,10,11)(13,15,19,22)(14,20,17,18)(16,24,27,23)(21,28,25,26);(1,2,4,3)(7,12)(9,11)(13,16,19,27)(14,21,17,25)(15,23,22,24)(18,28,20,26);(6,10)(9,11)(13,17,19,14)(15,20,22,18)(16,25,27,21)(23,26,24,28);(1,3,4,2)(14,17)(15,22)(21,25)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717782);