Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,20,33)(8,22)(9,14)(10,15)(11,18,30,24)(12,19,31,21)(13,25)(16,27)(23,34,36,29)(26,32)(28,35);(5,7)(6,11)(8,23)(9,19)(10,20)(12,13)(14,29)(15,30)(16,34)(17,32)(18,35)(21,27)(22,33)(24,25)(26,36)(28,31);(2,4)(5,7)(6,12)(8,24)(9,21)(10,20)(11,28)(13,30)(14,33)(15,31)(16,17)(18,26)(19,27)(22,34)(23,25)(29,32)(35,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231480, #72231535, #72231546, … |
| maximal normal subgroup of | #72231480, #72231535, #72231546, … |
| maximal quotient of | #72264823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54852);