Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
536 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)(5,6,9,13,11,16,10,15)(7,17,19,30,21,34,20,33)(8,22,25,32,24,31,26,14)(12,23,28,18,29,36,27,35);(5,7,9,20,11,21,10,19)(6,12,15,28,16,29,13,27)(8,23,25,35,24,36,26,18)(14,30,31,34,32,33,22,17);(2,4)(6,13,16,15)(8,24)(12,27,29,28)(14,31,32,22)(17,30,34,33)(18,35)(23,36)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320963, #72320995, #72321042, … |
| maximal normal subgroup of | #72320963, #72320995, #72321042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54442);