Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 |
| prime radical | #37649 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2236 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)(5,6)(7,10)(8,11)(9,12)(13,14,18,21)(15,22,23,28)(16,20,17,26)(19,25,27,24);(5,7)(6,9)(8,12)(10,11)(13,15,16,25,18,23,17,24)(14,19,26,28,21,27,20,22);(2,4)(7,12)(9,10)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,27)(25,28);(5,7)(6,10)(8,12)(9,11)(15,23)(19,27)(22,28)(24,25);(7,12)(9,10)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,28,27,22)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504365);