Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 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)(5,6,9,16,11,19,28,15,12,20,10,29,25,18,30,17)(7,21,22,13,23,31,36,34,24,32,8,14,26,35,27,33);(2,4)(5,7,10,22,11,23,30,36,12,24,9,8,25,26,28,27)(6,13,17,32,18,33,29,31,20,14,15,21,19,34,16,35);(5,8,9,26,11,27,28,7,12,22,10,23,25,36,30,24)(6,14,16,35,19,33,15,21,20,13,29,31,18,34,17,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cg,e^b=ei,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216223, C4×#12068, #72223666, … |
| maximal normal subgroup of | #72216223, C4×#12068, #72223666, … |
| maximal quotient of | #72223664, C2×#128694, #72242119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35890);