Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153167 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| 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 | |
578 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 578 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:
(4,16)(5,18)(7,21)(9,25)(10,19)(11,27)(12,14)(13,31)(17,20)(24,26)(28,29)(30,32);(1,2)(3,11)(4,9)(5,19)(6,8)(7,13)(10,18)(12,29)(14,27)(15,28)(16,26)(17,24)(20,25)(21,32)(22,31)(23,30);(1,3)(2,7)(4,13)(5,12)(6,15)(8,21)(9,11)(10,22)(14,18)(16,31)(17,32)(19,23)(20,30)(24,28)(25,29)(26,27)Polycyclic presentation:
2^9|d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dhi,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=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72286052, #72286072 |
| maximal normal subgroup of | #72286052, #72286072 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60363);