Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10)(11,17,18,23)(12,15)(13,16)(14,19,22,20)(21,24);(1,3)(2,5)(4,6)(7,8)(14,22)(17,23);(9,11)(10,14)(12,20)(13,18)(15,23)(16,22)(17,24)(19,21)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,d^c=di,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238761, #72238762, #72238772, … |
| maximal normal subgroup of | #72238761, #72238762, #72238772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22579);