Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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×#3280 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2185 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,25)(17,24)(19,26)(20,23)(21,27)(22,28);(1,2)(3,4)(6,9)(10,12)(13,15,20,26)(14,18,23,19)(16,21,17,22)(24,28,25,27);(13,16)(14,21)(15,24)(17,20)(18,27)(19,28)(22,23)(25,26);(2,4)(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,22,23,21)(15,25,26,24)(18,28,19,27);(6,9)(10,12)(13,18,20,19)(14,15,23,26)(16,27,17,28)(21,24,22,25)Polycyclic presentation:
2^9|b^2=i,d^2=fi,e^2=i,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,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424027);