Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60423 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,15,23,17,28,16,8)(4,9,19,25,20,27,21,26)(14,24,18,32,31,29,30,22);(1,3)(2,8)(4,18)(5,15)(6,16)(7,17)(9,24)(10,28)(11,23)(12,13)(14,21)(19,31)(20,30)(22,26)(25,32)(27,29);(1,4)(2,9,11,27)(3,14)(5,21)(6,19)(7,20)(8,22,23,32)(10,25,12,26)(13,29,28,24)(15,30)(16,18)(17,31);(2,10)(3,15)(5,6)(8,23)(9,25)(11,12)(14,30)(16,17)(18,31)(19,21)(22,32)(26,27)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216130, #72216181, … |
| maximal normal subgroup of | #72216013, #72216130, #72216181, … |
| maximal quotient of | C2×#385937, #72252162, #72255740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293133);