Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | #33558 | C2 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
503 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8)(3,11)(5,18,16,26)(6,20,17,10)(7,15)(9,22,24,19)(12,29)(13,23)(14,31)(21,25)(27,30)(28,32);(1,2)(3,12)(4,8)(5,9)(6,20)(7,14)(10,17)(11,27)(13,31)(15,28)(16,24)(18,22)(19,26)(21,30)(23,29)(25,32);(1,3)(2,7)(4,15)(5,14)(6,21)(8,11)(9,12)(10,25)(13,17)(16,27)(18,30)(19,32)(20,23)(22,29)(24,28)(26,31)Polycyclic presentation:
2^9|a^2=h,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72286410 |
| maximal normal subgroup of | #72286410 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60401);