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 | #36434 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14,19,24)(15,20,23,18)(16,22,27,28)(17,21,26,25);(1,2)(3,4)(6,10)(11,12)(13,15,19,23)(14,18,24,20)(16,25,27,21)(17,22,26,28);(13,16,19,27)(14,21,24,25)(15,22,23,28)(17,18,26,20);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,22)(15,25)(16,18)(19,26)(20,27)(21,23)(24,28);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,23,24,15)(16,26,27,17)(21,28,25,22)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,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, 7429137);