Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394133 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3934 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)(3,14)(4,9)(5,12)(6,11)(7,13)(8,17)(10,19)(15,29)(16,23)(18,25)(20,28)(21,24)(22,27)(26,30)(31,32);(1,3)(2,8)(4,16)(5,21)(6,17)(7,18)(9,26)(10,23)(11,25)(12,14)(13,24)(15,19)(20,31)(22,30)(27,32)(28,29);(1,4)(2,9)(3,15)(5,22)(6,19)(7,20)(8,23)(10,11)(12,27)(13,28)(14,26)(16,21)(17,31)(18,30)(24,32)(25,29);(1,4)(2,10)(3,16)(5,19)(6,22)(7,20)(8,23)(9,11)(12,28)(13,27)(14,29)(15,21)(17,30)(18,31)(24,32)(25,26)Polycyclic presentation:
2^9|e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216013, #72216112, … |
| maximal normal subgroup of | #72215995, #72216013, #72216112, … |
| maximal quotient of | #72237177, #72237512, #72237513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301329);