Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #117 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1770 |
| prime radical | C24 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
592 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)(5,6,9,16,11,15,24,33,12,17,10,26,8,13,23,28)(7,18,20,34,21,31,36,32,22,27,25,35,19,14,29,30);(2,4)(6,13,17,15)(14,27,31,18)(16,28,26,33)(30,35,32,34);(5,7,10,20,11,21,23,36,12,22,9,25,8,19,24,29)(6,14,16,32,15,18,33,35,17,31,26,30,13,27,28,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fi,g^a=gi,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72314212, #72314225, #72314229, … |
| maximal normal subgroup of | #72314212, #72314225, #72314229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53519);