Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,9,7,8,10,12,11)(13,14,16,21,18,22,17,27)(15,23,25,28,26,19,24,20);(5,7,9,10,8,11,12,6)(14,19,22,23)(15,24)(16,17)(20,21,28,27)(25,26);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216306, #72216334, #72216338, … |
| maximal normal subgroup of | #72216306, #72216334, #72216338, … |
| maximal quotient of | #72216304, #72216306, #72225350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47584);