Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405494 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14913 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(5,17)(8,21)(9,19)(10,20)(12,18)(15,28)(22,29)(23,30)(26,31)(27,32);(1,2,5,7,6,10,17,20)(3,12,25,18,11,8,14,21)(4,9,15,19,16,23,28,30)(13,26,32,29,24,22,27,31);(1,3,5,14,6,11,17,25)(2,8,7,18,10,12,20,21)(4,13,15,27,16,24,28,32)(9,22,19,29,23,26,30,31);(1,4)(2,9)(3,13)(5,15)(6,16)(7,19)(8,22)(10,23)(11,24)(12,26)(14,27)(17,28)(18,29)(20,30)(21,31)(25,32)Polycyclic presentation:
2^9|b^2=eh,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319375, #72319399, #72319453 |
| maximal normal subgroup of | #72319375, #72319399, #72319453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312690);