Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153305 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13424 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,12,16,7)(4,17)(5,10)(6,9)(8,18)(11,20)(13,24,29,31)(14,22,15,23)(19,26)(21,28,27,30)(25,32);(1,3)(2,7)(4,13)(5,14)(6,16)(8,24)(9,12)(10,23)(11,22)(15,20)(17,31)(18,29)(19,30)(21,25)(26,27)(28,32);(1,4)(2,8,9,17)(3,13)(5,19)(6,18)(7,21,12,27)(10,25,11,26)(14,28)(15,30)(16,29)(20,32)(22,31,23,24)Polycyclic presentation:
2^9|a^2=h,c^2=d,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,e^b=eg,g^b=gi,h^b=hi,e^c=ehi,g^c=gh,f^e=fh,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72216577, #72216589, #72216601, … |
| maximal normal subgroup of | #72216577, #72216589, #72216601, … |
| maximal quotient of | #72250911, #72250914, C2×#153305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60501);