Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1864 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,18,21,35)(8,23)(9,26)(10,29)(11,16)(12,19,32,27)(13,20,33,36)(14,24)(15,25)(17,28)(22,34,30,31);(2,4)(5,7)(6,12,16,32)(8,20)(9,27)(10,30)(11,21)(13,29,33,17)(14,31,26,34)(15,35,23,18)(19,24)(22,28)(25,36);(5,7)(6,13)(8,20)(9,27)(10,30)(11,21)(12,29)(14,18)(15,31)(16,33)(17,32)(19,24)(22,28)(23,34)(25,36)(26,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216636, #72250533, #72250659, … |
| maximal normal subgroup of | #72216636, #72250533, #72250659, … |
| maximal quotient of | #72216578 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53933);