Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,10,13,24,16)(7,17,19,32,21,11,20,29)(8,22,25,14,26,27,34,31)(12,23,28,36,30,35,33,18);(1,2)(3,4)(5,7)(6,11)(8,18)(9,20)(10,21)(12,27)(13,17)(14,28)(15,32)(16,29)(19,24)(22,30)(23,34)(25,35)(26,36)(31,33);(2,4)(5,7)(6,12,13,30)(8,23)(9,19)(10,21)(11,27,17,22)(14,29,31,32)(15,28,16,33)(18,34)(20,24)(25,36)(26,35)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^b=gi,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250403, #72250416, #72250418, … |
| maximal normal subgroup of | #72250403, #72250416, #72250418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55804);