Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,24,22,27,21,26)(19,28,29,32,20,25,30,31);(2,5)(6,8)(18,24)(19,29)(20,30)(21,23)(25,28)(26,27);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,21,30,22,20,23,29)(18,25,26,32,27,28,24,31);(17,20,21,29,22,19,23,30)(18,25,26,32,27,28,24,31);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=fi,c^2=f,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72343709, #72343710, #72343714, … |
| maximal normal subgroup of | #72343709, #72343710, #72343714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372033);