Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(13,14)(15,19,20,17)(16,18);(2,4)(5,6)(7,8)(9,10,11,12)(13,15)(14,17)(16,20)(18,19);(6,8)(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72219233, #72225649, #72225659, … |
| maximal normal subgroup of | #72219233, #72225649, #72225659, … |
| maximal quotient of | #72219229, #72219230, #72309370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30716);