Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395587 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2411 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14918 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1797 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)(4,16)(5,21,23,10)(6,12)(7,11)(8,13)(9,19)(15,29,25,28)(17,20)(18,30,32,31)(22,26,24,27);(1,3)(2,4,13,20)(5,15)(6,16)(7,17)(8,9)(10,18,21,32)(11,14,12,19)(22,30)(23,25)(24,31)(26,28,27,29);(1,4)(2,9)(3,13)(5,18)(6,19)(7,14)(8,20)(10,25)(11,16)(12,17)(15,21)(22,29)(23,32)(24,28)(26,30)(27,31);(1,5)(2,10)(3,15)(4,18)(6,22)(7,24)(8,23)(9,25)(11,26)(12,27)(13,21)(14,28)(16,30)(17,31)(19,29)(20,32)Polycyclic presentation:
2^9|a^2=ef,b^2=h,e^2=i,f^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,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319352, #72319406, #72319444 |
| maximal normal subgroup of | #72319352, #72319406, #72319444 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312966);