Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #96924 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,15,18,16,17,19,20)(22,26,23,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,17)(15,19)(21,22,25,23)(24,27,28,26);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(15,19)(21,23,25,22)(24,26,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=i,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,e^c=ei,g^c=ghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28826);