Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417718 |
| 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 | |
2401 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)(4,18)(5,12)(6,10)(7,11)(8,16)(9,20)(14,23)(15,24)(17,29)(19,28)(21,27)(22,25)(26,31)(30,32);(1,3)(2,8)(4,14)(5,22)(6,16)(7,17)(9,26)(10,29)(11,25)(12,13)(15,19)(18,32)(20,30)(21,31)(23,28)(24,27);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,23)(10,27)(11,28)(12,18)(13,24)(15,22)(16,30)(17,31)(25,26)(29,32);(1,4)(2,9)(3,15)(5,20)(6,19)(7,21)(8,24)(10,18)(11,28)(12,27)(13,23)(14,22)(16,31)(17,30)(25,32)(26,29)Polycyclic presentation:
2^9|e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,e^d=ei,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215981, #72216059, #72216154, … |
| maximal normal subgroup of | #72215981, #72216059, #72216154, … |
| maximal quotient of | C2×#417718, #72255869, #72265363 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324914);