Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15,18,11)(8,12)(9,21,22,13)(10,14)(16,23,30,25)(17,24)(19,32,33,26)(20,27)(28,31)(29,35)(34,36);(1,2,4,3)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,24)(14,15)(19,29)(20,30)(21,28)(22,31)(23,27)(26,35)(32,34)(33,36);(5,8,10,20)(6,12,14,27)(7,16,18,30)(9,19,22,33)(11,23,15,25)(13,26,21,32)(17,29,31,36)(24,34,28,35);(1,3,4,2)(5,9)(6,13)(7,17)(8,19)(10,22)(11,24)(12,26)(14,21)(15,28)(16,29)(18,31)(20,33)(23,34)(25,35)(27,32)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,d^2=eg,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284354, #72284459, #72284786, … |
| maximal normal subgroup of | #72284354, #72284459, #72284786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271370);