Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,17,12,18,11,28)(7,19,22,30,24,33,23,29)(8,16,25,34,27,13,9,15)(14,20,31,21,32,36,35,26);(1,2)(3,4)(6,13)(8,25)(9,27)(10,11)(14,29)(15,28)(16,18)(17,34)(19,35)(20,21)(22,23)(26,36)(30,32)(31,33);(2,4)(5,7)(6,14)(8,26)(9,20)(10,23)(11,22)(12,24)(13,29)(15,33)(16,30)(17,35)(18,32)(19,34)(21,27)(25,36)(28,31)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72224936, #72226822, … |
| maximal normal subgroup of | #72224903, #72224936, #72226822, … |
| maximal quotient of | #72224900, #72238787, C2×#147894 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55090);