Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464757 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16509 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1511 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)(3,12,15,7)(4,8)(5,19)(6,10)(9,18)(11,21)(13,22,31,28)(14,23,32,25)(16,24,30,29)(17,27)(20,26);(1,3,5,16,6,15,21,30)(2,7,19,29,10,12,11,24)(4,14,17,31,18,32,20,13)(8,25,27,22,9,23,26,28);(1,4)(2,8)(3,13)(5,20)(6,18)(7,22)(9,10)(11,27)(12,28)(14,30)(15,31)(16,32)(17,21)(19,26)(23,24)(25,29);(1,4)(2,9,10,8)(3,14)(5,17)(6,18)(7,23,12,25)(11,27,19,26)(13,30)(15,32)(16,31)(20,21)(22,29,28,24)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72240432, #72240437, #72240448, … |
| maximal normal subgroup of | #72240432, #72240437, #72240448, … |
| maximal quotient of | #72251294, #72255620, C2×#464757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371953);