Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14428 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
969 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)(3,12,15,26)(4,17,19,24)(5,9)(6,11)(7,16,23,13)(8,18,25,20)(10,21)(14,29)(22,31)(27,28)(30,32);(3,13,15,16)(7,12,23,26)(14,30,31,28)(22,32,29,27);(1,3,6,15)(2,7)(4,14,19,31)(5,16,21,13)(8,22)(9,12)(10,26)(11,23)(17,32)(18,28,20,30)(24,27)(25,29);(1,4)(2,8)(3,14)(5,20)(6,19)(7,22)(9,24)(10,17)(11,25)(12,27)(13,28)(15,31)(16,30)(18,21)(23,29)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284930, #72287280, #72287341, … |
| maximal normal subgroup of | #72284930, #72287280, #72287341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293606);