Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,21,16,20)(15,23,26,27,24,28,25,19);(2,3)(5,6)(7,11)(8,10)(9,12)(15,24)(19,27)(23,28)(25,26);(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,19,21,27)(16,26,17,25)(20,23,22,28);(1,3,4,2)(7,12)(9,11)(13,16,18,17)(14,20,21,22)(15,25,24,26)(19,28,27,23);(6,10)(9,11)(14,21)(19,27)(20,22)(23,28)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913433);