Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#99 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| socle | C2 | C23×#99 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,9)(5,16,17,7)(6,10)(11,20)(12,21)(13,26,24,18)(14,27,28,19)(15,22)(23,30)(25,32,31,29);(2,7)(8,18)(9,19)(10,16)(20,26)(21,29)(22,27)(30,32);(1,3)(2,8)(4,12)(5,13)(6,11)(7,18)(9,21)(10,20)(14,25)(15,23)(16,26)(17,24)(19,29)(22,30)(27,32)(28,31);(3,11)(8,20)(12,23)(13,24)(18,26)(21,30)(25,31)(29,32);(1,4,6,15)(2,9,10,22)(3,12,11,23)(5,14,17,28)(7,19,16,27)(8,21,20,30)(13,25,24,31)(18,29,26,32)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=bf,f^a=fh,h^a=hi,g^b=gh,d^c=di,g^f=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237528, #72237542, #72237549, … |
| maximal normal subgroup of | #72237528, #72237542, #72237549, … |
| maximal quotient of | #72302947, #72302949, C2×#7622450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529646);