Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383160 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14357 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13)(4,17)(5,8,6,11)(9,21)(10,20)(14,27,29,32)(15,25)(16,23)(18,22)(19,26)(24,30,31,28);(2,8)(4,18)(5,6)(9,23)(10,17)(11,12)(13,25)(14,28)(15,21)(19,20)(27,32)(29,30);(1,3)(2,9)(4,14)(5,15)(6,21)(7,16)(8,13)(10,24)(11,23)(12,25)(17,31)(18,28)(19,30)(20,29)(22,27)(26,32);(1,4)(2,10)(3,14)(5,19)(6,18)(7,20)(8,22)(9,24)(11,26)(12,17)(13,27)(15,30)(16,29)(21,28)(23,32)(25,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216181, #72216192, … |
| maximal normal subgroup of | #72215958, #72216181, #72216192, … |
| maximal quotient of | C2×#383160, #72238761, #72238762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290356);