Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14389 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,20,27)(3,11,14,22,15,26,13,7)(4,16,18,24,19,8,17,25)(12,28,29,21,31,32,30,23);(1,3)(2,7)(4,12)(5,13)(6,15)(8,23)(9,26)(10,22)(11,27)(14,20)(16,21)(17,29)(18,30)(19,31)(24,28)(25,32);(1,4,5,17,6,19,20,18)(2,8,9,24,10,16,27,25)(3,12,14,30,15,31,13,29)(7,21,11,28,22,23,26,32);(2,9,10,27)(3,13,15,14)(7,22)(8,24,16,25)(11,26)(12,29,31,30)(21,23)(28,32)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319342, #72319366, #72319453 |
| maximal normal subgroup of | #72319342, #72319366, #72319453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291972);