Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,11)(9,10)(13,14)(15,23)(16,19)(17,27)(18,20)(21,25)(22,24)(26,28);(1,2)(3,4)(5,7,12,6,11,9,10,8)(13,15,24,26,18,25,17,16)(14,19,27,21,20,28,22,23);(2,4)(5,8,12,7,11,6,10,9)(14,20)(15,25)(16,26)(22,27);(5,9,12,8,11,7,10,6)(13,16,17,25,18,26,24,15)(14,21,22,19,20,23,27,28)Polycyclic presentation:
2^9|b^2=eh,c^2=fh,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 205893);