Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #401 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,19,14)(12,15,21,24)(13,16)(18,23)(20,22);(5,6)(7,8)(9,11)(10,14)(12,20,21,18)(13,19)(15,23,24,22)(16,17);(2,4)(6,8)(9,12)(10,15,16,24)(11,18)(13,21)(14,22,17,23)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237344, #72265862, #72265865, … |
| maximal normal subgroup of | #72237344, #72265862, #72265865, … |
| maximal quotient of | #72237346, #72266076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30980);