Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(14,18,19,20)(21,22,23,25,24,26,27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,20)(17,19);(2,4)(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279383, #72279393, #72279399, … |
| maximal normal subgroup of | #72279383, #72279393, #72279399, … |
| maximal quotient of | #72279883, #72280156, #72280159 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4111);