Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×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 | |
1846 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 1846 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,22);(2,6)(3,4)(7,8)(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,23)(22,24);(2,7)(3,4)(6,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,19)(17,24)(18,23);(10,17)(15,18)(16,24)(19,23)Polycyclic presentation:
2^9|e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214488, #72214656, #72219621, … |
| maximal normal subgroup of | #72214488, #72214656, #72219621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141085);