Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12,14,30)(7,16,18,17)(8,20)(9,24)(11,23)(13,31)(15,21,27,25)(19,22,28,26)(29,36)(32,35);(1,2,3,4)(5,6,10,14)(7,13,18,31)(8,21,9,25)(11,27,23,15)(12,20,30,24)(16,29,17,35)(19,34,28,33)(22,36,26,32);(5,7,11,28,10,18,23,19)(6,13,27,33,14,31,15,34)(8,22,20,17,9,26,24,16)(12,29,21,36,30,35,25,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250147, #72250150, #72250158, … |
| maximal normal subgroup of | #72250147, #72250150, #72250158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54493);