Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14724 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3463 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)(3,14,17,8)(4,20,21,9)(5,12)(6,13)(7,11)(10,22,28,23)(15,29)(16,31)(18,26,19,25)(24,30)(27,32);(1,3)(2,8)(4,16)(5,19)(6,18)(7,17)(9,27)(10,24)(11,14)(12,26)(13,25)(15,22)(20,31)(21,32)(23,30)(28,29);(1,4)(2,9)(3,15)(5,23)(6,22)(7,21)(8,24)(10,13)(11,20)(12,28)(14,29)(16,19)(17,30)(18,32)(25,31)(26,27);(1,4)(2,10)(3,16)(5,22)(6,23)(7,21)(8,24)(9,13)(11,28)(12,20)(14,29)(15,19)(17,32)(18,30)(25,27)(26,31)Polycyclic presentation:
2^9|a^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72216059, #72216141, … |
| maximal normal subgroup of | #72215974, #72216059, #72216141, … |
| maximal quotient of | #72237178, C2×#395471, #72252197 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302667);