Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33520 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1055 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)(3,4)(6,11,14,26)(8,19)(9,22)(12,24,28,35)(13,25,15,20)(17,32)(18,31)(21,23)(27,33,29,36)(30,34);(5,6)(7,12)(8,20)(9,13)(10,14)(11,23)(15,22)(16,28)(17,33)(18,29)(19,26)(21,25)(24,34)(27,31)(30,36)(32,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,24)(13,29)(14,28)(15,27)(19,32)(20,33)(21,30)(22,31)(23,34)(25,36)(26,35);(6,13,14,15)(7,16)(8,21)(11,25,26,20)(12,27,28,29)(18,31)(19,23)(24,33,35,36)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225127, #72250867, #72250872, … |
| maximal normal subgroup of | #72225127, #72250867, #72250872, … |
| maximal quotient of | #72225063, #72225064, #72286138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402919);