Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232905 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1462 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,20)(21,23)(22,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,23)(17,19)(18,24);(1,2)(3,6)(4,7)(5,8)(10,16)(12,13)(15,19)(17,18)(20,22)(23,24);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,23,24,19)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338241, #72338246, #72338336, … |
| maximal normal subgroup of | #72338241, #72338246, #72338336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140102);