Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2544 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14806 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1420 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)(3,14)(4,18)(5,13)(6,12)(7,11)(8,16)(9,17)(10,19)(15,29)(20,26)(21,27)(22,24)(23,32)(25,30)(28,31);(1,3,7,22,6,17,5,16)(2,8,13,9,12,24,11,14)(4,15,19,30,20,32,21,31)(10,23,18,25,27,29,26,28);(1,3)(2,9)(4,15)(5,16)(6,17)(7,22)(8,11)(10,25)(12,14)(13,24)(18,29)(19,30)(20,32)(21,31)(23,26)(27,28);(1,4,7,19,6,20,5,21)(2,10,13,18,12,27,11,26)(3,15,22,30,17,32,16,31)(8,23,9,25,24,29,14,28)Polycyclic presentation:
2^9|b^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319466 |
| maximal normal subgroup of | #72319466 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307884);