Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2052 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)(5,6)(7,10,17,25)(8,12)(9,21)(11,14)(13,24)(15,19,30,35)(16,31,33,23)(18,28)(20,27)(22,26)(29,36,32,34);(5,7,10,14,11,17,25,6)(8,15,19,27,20,30,35,12)(9,16,23,21,24,33,31,13)(18,29,34,28,22,32,36,26);(2,4)(5,8,11,20)(6,12,14,27)(7,15,17,30)(9,22,24,18)(10,19,25,35)(13,28,21,26)(16,32,33,29)(23,36,31,34);(5,9,11,24)(6,13,14,21)(7,16,17,33)(8,18,20,22)(10,23,25,31)(12,26,27,28)(15,29,30,32)(19,34,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311264, #72311463, #72311505, … |
| maximal normal subgroup of | #72311264, #72311463, #72311505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401885);