Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| 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)(8,15)(9,14)(10,27)(11,16)(12,19)(13,20)(17,23)(21,29)(22,34)(24,35)(25,30)(26,33)(28,31)(32,36);(5,7)(6,12,16,30)(8,20)(9,25)(10,28)(11,22)(13,17,32,27)(14,29,33,31)(15,34,35,18)(19,26)(21,23)(24,36);(5,7)(6,13)(8,20)(9,25)(10,28)(11,22)(12,17)(14,18)(15,29)(16,32)(19,26)(21,23)(24,36)(27,30)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=ehi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216636, #72281148, #72281194, … |
| maximal normal subgroup of | #72216636, #72281148, #72281194, … |
| maximal quotient of | #72216515 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53943);