Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,13,11,16,9,15)(7,17,21,28,19,12,20,26)(8,14,24,29,25,32,23,31)(18,30,34,36,22,27,33,35);(5,7)(6,12,16,17)(8,22,25,18)(9,20)(10,21)(11,19)(13,26,15,28)(14,30)(23,34,24,33)(27,32)(29,36)(31,35);(1,2,4,3)(6,13)(9,10)(12,26)(14,29)(15,16)(17,28)(20,21)(23,24)(27,35)(30,36)(31,32)(33,34);(1,3,4,2)(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,27)(13,29)(15,31)(16,32)(17,30)(19,22)(20,33)(21,34)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,f^2=i,g^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,f^c=fi,g^c=gi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280827, #72281002, #72281219, … |
| maximal normal subgroup of | #72280827, #72281002, #72281219, … |
| maximal quotient of | #72313509 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264159);