Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417702 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2413 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)(3,13)(4,19)(5,11)(6,10)(7,12)(8,16)(9,22)(14,31)(15,24)(17,26)(18,25)(20,30)(21,29)(23,28)(27,32);(1,3)(2,8)(4,20)(5,17)(6,16)(7,18)(9,27)(10,25)(11,13)(12,26)(14,22)(15,21)(19,31)(23,32)(24,28)(29,30);(1,4)(2,9)(3,14)(5,21)(6,22)(7,23)(8,24)(10,28)(11,19)(12,29)(13,27)(15,18)(16,32)(17,20)(25,30)(26,31);(1,4)(2,9)(3,15)(5,22)(6,21)(7,23)(8,24)(10,19)(11,28)(12,29)(13,30)(14,18)(16,32)(17,20)(25,27)(26,31)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72215968, #72215981, … |
| maximal normal subgroup of | #72215906, #72215968, #72215981, … |
| maximal quotient of | C2×#417702 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324898);