Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #271643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(7,12)(9,11)(13,14,18,23)(15,21,26,20)(16,27,25,28)(17,24,19,22);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,15,18,26)(14,20,23,21)(16,19,25,17)(22,27,24,28);(6,9,10,11)(13,16,19,26,18,25,17,15)(14,21,24,28,23,20,22,27);(1,3,4,2)(13,17,18,19)(14,22,23,24)(15,25,26,16)(20,28,21,27)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 205152);