Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,12)(10,15)(11,16)(13,19)(18,31)(20,25)(21,30)(22,29)(23,26)(24,28)(27,33)(32,36)(34,35);(6,12,16,15)(8,21,22,23)(13,25,28,17)(14,29)(18,32,34,33)(26,30)(27,36)(31,35);(1,2,4,3)(5,7)(6,13)(8,18)(9,19)(10,24)(11,20)(12,25)(14,27)(15,17)(16,28)(21,32)(22,34)(23,33)(26,35)(29,36)(30,31);(1,3,4,2)(5,8)(6,14)(7,18)(9,23)(10,21)(11,22)(12,26)(13,27)(15,30)(16,29)(17,31)(19,33)(20,34)(24,32)(25,35)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,f^c=fi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224693, #72312221, #72312223 |
| maximal normal subgroup of | #72224693, #72312221, #72312223 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263870);