Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128067 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,11,13,23)(4,8,16,22)(6,10,18,24)(7,17,20,14)(12,19,27,30)(15,21,28,31)(25,29,32,26);(1,3)(2,7,10,11)(4,12)(5,17,18,13)(6,14)(8,19,21,25)(9,23)(15,26)(16,29,28,27)(20,24)(22,32)(30,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,25)(13,27)(14,26)(17,29)(18,28)(20,30)(23,32)(24,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,b^a=bd,c^a=cg,d^a=dg,f^a=fh,h^a=hi,d^b=dh,e^b=eg,g^b=gi,d^c=di,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72255657, #72265901, #72265906, … |
| maximal normal subgroup of | #72255657, #72265901, #72265906, … |
| maximal quotient of | #72236625 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35263);