Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10,11,12)(14,18)(15,21)(16,23)(19,22)(20,26)(24,27);(1,2,3,4)(10,12)(13,14,16,18,17,20,23,26)(15,22,21,28,24,19,27,25);(13,15,23,27,17,24,16,21)(14,19,26,28,20,22,18,25);(6,8)(14,20)(18,26)(19,22)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264578, #72264595, #72264642, … |
| maximal normal subgroup of | #72264578, #72264595, #72264642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300195);