Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,16,12,14,26,29,10,15,24,30,9,22,23,35)(7,17,19,32,20,28,25,31,18,33,34,36,8,13,21,27);(1,2,4,3)(5,7,9,8,10,18,12,20)(6,13,14,17,15,28,22,33)(11,19,23,21,24,34,26,25)(16,27,29,32,30,31,35,36);(1,3,4,2)(5,8,9,18,10,20,12,7)(6,13,14,17,15,28,22,33)(11,25,23,19,24,21,26,34)(16,31,29,36,30,27,35,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239416, #72239421, #72239430, … |
| maximal normal subgroup of | #72239416, #72239421, #72239430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55878);