Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #289819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22,25,28,26,30,24,29)(23,27,33,36,32,34,31,35);(5,7,9,16,10,17,8,15)(6,11,12,14,13,19,18,20)(21,23,25,33,26,32,24,31)(22,27,28,36,30,34,29,35);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(22,28)(23,31)(24,25)(29,30)(32,33)(35,36);(5,7,8,15,10,17,9,16)(6,11,18,20,13,19,12,14)(23,32)(27,34)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=fi,b^2=eg,c^2=h,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 230270);