Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188049 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10)(3,9,12,23)(5,7,14,19)(6,8,15,21)(11,18,24,30)(13,20,27,31)(16,17,28,29)(22,26,32,25);(2,7)(4,14)(8,17)(9,18)(12,24)(15,28)(20,22)(26,27);(2,8)(3,11)(4,15)(7,17)(9,22)(12,26)(13,25)(14,28)(18,20)(23,30)(24,27)(31,32);(1,3,6,13)(2,9,8,20)(4,12,15,27)(5,11,16,25)(7,18,17,22)(10,23,21,31)(14,24,28,26)(19,30,29,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266963, #72266966, #72266968, … |
| maximal normal subgroup of | #72266963, #72266966, #72266968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95254);