Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94441 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7307 |
| commutator | C2×C42 | C42 |
| Frattini, prime radical, cosocle | C2×#535 | C22 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
496 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 2 generators on 32 points:
(1,2,4,9)(3,12,14,7)(5,10,19,8)(6,11,17,26)(13,28,15,24)(16,27,30,22)(18,23,20,25)(21,31,32,29);(1,3)(2,7)(4,13,17,29)(5,15)(6,16)(8,12,23,27)(9,24,26,32)(10,21)(11,22)(14,19,30,18)(20,31)(25,28)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,f^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,h^a=hi,c^b=cg,d^b=dfh,g^b=gi,d^c=dhi,e^c=ei,e^d=ei,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72213646, #72213678, #72213695, … |
| maximal normal subgroup of | #72213646, #72213678, #72213695, … |
| maximal quotient of | #72338270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1637);