Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99899 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(5,6,9,12)(7,11,8,10)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(2,4)(6,10)(7,8)(11,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314152 |
| maximal normal subgroup of | #72314152 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21708);