Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16369 |
| 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 | |
933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13)(3,14)(4,9)(5,11,7,12)(8,19)(10,20)(15,27,30,32)(16,24,31,23)(17,26)(18,25)(21,29)(22,28);(1,3,6,18)(2,8)(4,16,21,31)(5,17,7,19)(9,27)(10,24)(11,14)(12,25)(13,26)(15,20,30,22)(23,28)(29,32);(1,4,5,20,6,21,7,22)(2,9,11,10,13,29,12,28)(3,15,17,16,18,30,19,31)(8,23,14,27,26,24,25,32);(1,4)(2,10)(3,16)(5,22)(6,21)(7,20)(8,24)(9,11)(12,29)(13,28)(14,27)(15,17)(18,31)(19,30)(23,26)(25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338619, #72338621, #72338629 |
| maximal normal subgroup of | #72338619, #72338621, #72338629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367860);