Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,15,11,16,9,13)(7,17,20,30,21,34,19,33)(8,22,26,31,24,29,25,14)(12,23,27,18,28,36,32,35);(5,7,10,19,11,21,9,20)(6,12,13,27,16,28,15,32)(8,23,26,35,24,36,25,18)(14,30,29,34,31,33,22,17);(1,2,3,4)(6,13,16,15)(8,24)(12,27,28,32)(14,22,31,29)(17,33,34,30)(18,35)(23,36)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,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 | #72321110, #72321156, #72321232, … |
| maximal normal subgroup of | #72321110, #72321156, #72321232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54663);