Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2011 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1152 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)(10,28)(11,16)(12,19)(13,20)(17,24)(21,30)(22,36)(23,33)(25,35)(26,31)(27,34)(29,32);(5,7)(6,12)(8,23)(9,19)(10,29)(11,22)(13,28)(14,32)(15,18)(16,31)(17,33)(20,25)(21,24)(26,27)(30,34)(35,36);(5,7)(6,13)(8,20)(9,26)(10,29)(11,22)(12,17)(14,18)(15,30)(16,33)(19,27)(21,24)(23,25)(28,31)(32,35)(34,36)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,h^b=hi,d^c=dhi,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231508, #72231547, #72231549, … |
| maximal normal subgroup of | #72231508, #72231547, #72231549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55072);