Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,12,7,13,5,10)(3,14,17,4,18,19,15,20)(8,11,25,30,26,31,23,29)(9,24,28,32,16,22,27,21);(1,3,8,16)(2,4,11,21)(5,15,25,27)(6,17,23,28)(7,18,26,9)(10,19,30,22)(12,14,29,24)(13,20,31,32);(1,4,7,20)(2,9,13,16)(3,11,18,31)(5,22,6,24)(8,21,26,32)(10,17,12,15)(14,25,19,23)(27,30,28,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237995, #72251842, #72252033, … |
| maximal normal subgroup of | #72237995, #72251842, #72252033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47025);