Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3048 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)(13,14,17,20,18,21,16,25)(15,22,23,27,24,19,26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,25)(18,21)(22,24)(23,27)(26,28);(2,4)(6,9)(10,12)(13,15)(14,19)(16,26)(17,23)(18,24)(20,28)(21,22)(25,27);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(7,11)(10,12)(15,24)(19,22)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=fgi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449871);