Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14345 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1192 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)(3,11,13,25)(4,8)(5,9)(6,10)(7,14,19,28)(12,26,15,29)(16,22)(17,23)(18,24)(20,27,21,30)(31,32);(7,19)(11,25)(20,21)(26,29);(1,3)(2,7,10,19)(4,15,17,12)(5,14)(6,13)(8,21)(9,11,24,25)(16,30,31,27)(18,28)(20,23)(22,29)(26,32);(1,4,5,16,6,17,18,31)(2,8,9,22,10,23,24,32)(3,12,14,27,13,15,28,30)(7,20,11,26,19,21,25,29)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=f,f^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216181, #72216238, … |
| maximal normal subgroup of | #72216037, #72216181, #72216238, … |
| maximal quotient of | C2×#382863, #72252164, #72256373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290059);