Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394043 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2613 |
| 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 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,4,7,11)(3,9,17,24,15,18,16,13)(5,10,20,26,21,19,22,25)(14,23,30,27,28,31,29,32);(1,3,8,15)(2,9)(4,18)(5,14,21,28)(6,16,7,17)(10,23)(11,24)(12,13)(19,31)(20,29,22,30)(25,27)(26,32);(1,4)(2,8)(3,13)(5,19)(6,11)(7,12)(9,17)(10,21)(14,27)(15,24)(16,18)(20,25)(22,26)(23,29)(28,32)(30,31);(1,5)(2,10)(3,14)(4,19)(6,22)(7,20)(8,21)(9,23)(11,26)(12,25)(13,27)(15,28)(16,30)(17,29)(18,31)(24,32)Polycyclic presentation:
2^9|a^2=ef,b^2=h,f^2=i,g^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,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319352, #72319406, #72319460 |
| maximal normal subgroup of | #72319352, #72319406, #72319460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312969);