Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396726 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14737 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1141 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)(4,14)(5,17)(8,21)(10,20)(12,28)(15,16)(18,24)(19,25)(23,32)(27,29);(1,2,5,7,6,10,17,20)(3,8,22,21,11,24,13,18)(4,9,15,19,16,26,14,25)(12,23,29,31,27,32,28,30);(1,3,5,13,6,11,17,22)(2,8,7,18,10,24,20,21)(4,12,15,28,16,27,14,29)(9,23,19,30,26,32,25,31);(1,4)(2,9)(3,12)(5,15)(6,16)(7,19)(8,23)(10,26)(11,27)(13,28)(14,17)(18,30)(20,25)(21,31)(22,29)(24,32)Polycyclic presentation:
2^9|b^2=ef,c^2=e,e^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,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338619, #72338625, #72338629 |
| maximal normal subgroup of | #72338619, #72338625, #72338629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303922);