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 | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(7,13)(8,14)(10,16)(12,17)(15,19)(18,20)(21,22,25,28)(23,26,30,35)(24,32,33,27)(29,36,31,34);(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,23)(22,26)(24,29)(25,30)(27,34)(28,35)(31,33)(32,36);(5,7,14,19,9,15,8,13)(6,10,17,20,11,18,12,16)(23,29,30,31)(26,34,35,36);(1,3,4,2)(6,11)(10,18)(12,17)(16,20)Polycyclic presentation:
2^9|a^2=i,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, 205163);