Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,27)(21,28)(22,26);(14,20)(15,23)(16,17)(21,22)(24,25)(27,28);(2,4)(5,7,9,8)(6,11,12,10)(14,21)(15,24)(16,17)(20,22)(23,25)(27,28);(5,8,9,7)(6,11,12,10)(15,25)(19,26)(23,24)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430355);