Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2154 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)(5,6,9,14,10,13,22,23)(7,15,18,29,17,11,21,25)(8,12,20,27,19,26,32,34)(16,28,31,36,30,24,33,35);(5,7,9,21)(6,11,23,29)(8,16,20,33)(10,17,22,18)(12,24,34,36)(13,15,14,25)(19,30,32,31)(26,28,27,35);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,26)(14,27)(15,28)(17,30)(18,31)(21,33)(22,32)(23,34)(25,35)(29,36);(8,19)(12,26)(16,30)(20,32)(24,28)(27,34)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,g^2=i,b^a=be,c^a=cf,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 | #72302930, #72302949, #72302965, … |
| maximal normal subgroup of | #72302930, #72302949, #72302965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401895);