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 | C2×#3306 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 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)(7,12)(8,10)(9,11)(13,14)(15,25)(16,20)(17,23)(18,22)(19,24)(21,26)(27,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,16)(17,18)(20,25)(21,28)(22,23)(26,27);(13,15)(14,20)(16,18)(17,27)(19,26)(21,22)(23,25)(24,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,17)(18,26)(19,27)(20,23)(22,28)(24,25);(6,11)(10,12)(15,26)(16,27)(20,28)(21,25)Polycyclic presentation:
2^9|b^2=f,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72284526, #72284714 |
| maximal normal subgroup of | #72284526, #72284714 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425200);