Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14357 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,12,7,10)(3,13,17,28,18,27,16,9)(4,19,22,5,23,25,21,26)(14,15,20,31,30,32,29,24);(1,3)(2,9)(4,20)(5,24)(6,17)(7,16)(8,18)(10,27)(11,13)(12,28)(14,21)(15,25)(19,32)(22,30)(23,29)(26,31);(1,4)(2,5)(3,14)(6,21)(7,22)(8,23)(9,15)(10,25)(11,19)(12,26)(13,24)(16,20)(17,29)(18,30)(27,31)(28,32);(1,5)(2,4)(3,15)(6,19)(7,25)(8,26)(9,14)(10,22)(11,21)(12,23)(13,29)(16,31)(17,24)(18,32)(20,27)(28,30)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,h^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,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318824, #72318876, #72318925, … |
| maximal normal subgroup of | #72318824, #72318876, #72318925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290370);