Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| 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 | |
638 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)(7,18,21,32)(8,22,20,13)(9,15)(10,25)(11,29)(12,17)(14,23,34,36)(16,27)(19,31,26,35)(24,30)(28,33);(1,2)(3,4)(5,7)(6,13)(8,11)(9,20)(10,26)(12,21)(14,33)(15,32)(16,35)(17,22)(18,29)(19,28)(23,24)(25,34)(27,36)(30,31);(2,4)(5,8,9,21,12,20,11,7)(6,14,15,31,17,34,29,35)(10,19,24,23,28,26,27,36)(13,30,18,33,22,16,32,25)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ehi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250819, #72250832, #72250838, … |
| maximal normal subgroup of | #72250819, #72250832, #72250838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55825);