Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151132 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12090 |
| commutator, Frattini, cosocle | #381 | C23 |
| LCS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 38 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:
(2,7)(3,11)(5,17)(8,23)(9,26)(10,22)(12,30)(13,18)(14,29)(15,19)(20,24)(21,28)(25,32)(27,31);(1,2)(3,8)(4,7)(5,9)(6,10)(11,24)(12,31)(13,25)(14,20)(15,26)(16,22)(17,28)(18,27)(19,21)(23,30)(29,32);(1,3)(2,8)(4,14)(5,18)(6,13)(7,20)(9,27)(10,25)(11,17)(12,16)(15,30)(19,29)(21,32)(22,31)(23,26)(24,28)Polycyclic presentation:
2^9|d^2=f,e^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,e^c=ei,f^c=fi,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72237823, #72237866, #72237867, … |
| maximal normal subgroup of | #72237823, #72237866, #72237867, … |
| maximal quotient of | #72226647, #72267775, #72267777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58328);