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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,9)(7,11,12,10)(13,14,17,18)(15,19,23,22)(16,24,25,20)(21,27,28,26);(2,3)(6,9)(10,11)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(1,2,4,3)(13,16,17,25)(14,20,18,24)(15,21,23,28)(19,26,22,27);(1,3,4,2)(14,18)(15,22)(19,23)(20,24)(21,27)(26,28);(5,7,8,12)(6,10,9,11)(15,23)(19,22)(21,28)(26,27)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469956);