Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149755 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11500 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
483 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 483 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,5,10)(3,7,13,19,12,20,16,29)(4,8,15,21)(6,9,17,23)(11,24,25,32,26,18,28,30)(14,22,27,31);(1,3)(2,7,9,20)(4,11)(5,16)(6,12)(8,18,22,24)(10,19,23,29)(13,17)(14,26)(15,28)(21,30,31,32)(25,27);(1,4)(2,8)(3,11)(5,15)(6,14)(7,18)(9,22)(10,21)(12,26)(13,25)(16,28)(17,27)(19,30)(20,24)(23,31)(29,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=ef,g^b=gi,d^c=di,e^d=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237425, #72237728, #72237731, … |
| maximal normal subgroup of | #72237425, #72237728, #72237731, … |
| maximal quotient of | #72338690 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56951);