Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187766 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,18,20,15)(12,16)(13,17)(14,19,23,21)(22,24);(2,6)(3,4)(9,11)(10,14)(12,21)(13,20)(15,16)(17,23)(18,24)(19,22);(9,11)(10,15)(12,19)(13,20)(14,16)(17,18)(21,22)(23,24);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=ghi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72305969, #72305972, #72305979, … |
| maximal normal subgroup of | #72305969, #72305972, #72305979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95020);