Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,20)(12,18)(13,17)(14,19)(15,22)(16,21)(23,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,24)(18,20);(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,19,18)(11,21,23,22)(15,24,16,20);(9,12,14,13)(10,18,19,17)(11,21,23,22)(15,20,16,24)Polycyclic presentation:
2^9|d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72338437, #72338438, #72338447, … |
| maximal normal subgroup of | #72338437, #72338438, #72338447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255313);