Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479598 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17217 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
858 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)(3,7)(4,14,16,8)(5,9)(6,10)(11,20,27,30)(12,28)(13,22)(15,24,18,23)(17,21)(19,25)(26,32,29,31);(1,3)(2,7)(4,11)(5,17,19,12)(6,13)(8,20)(9,21,25,28)(10,22)(14,30)(15,26,18,29)(16,27)(23,32,24,31);(1,4)(2,8)(3,11)(5,18,19,15)(6,16)(7,20)(9,24,25,23)(10,14)(12,29,17,26)(13,27)(21,32,28,31)(22,30);(1,5)(2,9)(3,12)(4,15)(6,19)(7,21)(8,23)(10,25)(11,26)(13,17)(14,24)(16,18)(20,31)(22,28)(27,29)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=g,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237952, #72237955, #72237957, … |
| maximal normal subgroup of | #72237952, #72237955, #72237957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386794);