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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1895 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)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,23,18)(16,25,27,22)(17,26,28,21);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,23)(14,18,24,20)(16,26,27,21)(17,22,28,25);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,25)(17,18)(19,27)(20,28)(22,23)(24,26);(2,4)(13,17,19,28)(14,22,24,25)(15,21,23,26)(16,18,27,20);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,23,24,15)(16,17,27,28)(21,22,26,25)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=i,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, 7427597);