Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2471 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15987 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,5,9,18,25)(3,11,27,21,13,7,14,29)(4,8,19,26,17,23,16,24)(12,28,15,32,30,20,31,22);(1,3,5,13)(2,7,9,11)(4,15)(6,14,18,27)(8,22)(10,21,25,29)(12,19)(16,30)(17,31)(20,26)(23,32)(24,28);(1,4)(2,8,9,23)(3,12)(5,17)(6,19)(7,20,11,28)(10,26,25,24)(13,30)(14,31)(15,27)(16,18)(21,32,29,22);(2,9)(8,23)(10,25)(24,26)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237073, #72237094, … |
| maximal normal subgroup of | #72237066, #72237073, #72237094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358370);