Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2308 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13784 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1431 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)(3,9,16,27)(4,19,20,23)(5,10,21,22)(6,13)(7,11)(8,12)(14,29)(15,25)(17,26,18,28)(24,32)(30,31);(1,3,8,18,6,16,7,17)(2,9,12,28,13,27,11,26)(4,14,10,25,20,30,22,32)(5,15,19,31,21,24,23,29);(1,4,7,22,6,20,8,10)(2,5,11,23,13,21,12,19)(3,14,18,25,16,30,17,32)(9,24,28,29,27,15,26,31);(1,5)(2,10)(3,15)(4,11)(6,21)(7,23)(8,19)(9,25)(12,20)(13,22)(14,26)(16,24)(17,29)(18,31)(27,32)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72338478, #72338483, #72338498, … |
| maximal normal subgroup of | #72338478, #72338483, #72338498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262681);