Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,10,13,25,29,11,14,24,30,8,15,22,31)(7,17,19,32,18,26,33,35,20,12,23,28,21,27,34,36);(1,2)(3,4)(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,27)(14,17)(15,26)(16,32)(19,24)(22,33)(25,34)(28,30)(29,35)(31,36);(2,4)(6,13,14,15)(9,24)(12,26,17,27)(16,31,30,29)(19,23)(22,25)(28,36,32,35)(33,34)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239415, #72239420, #72239426, … |
| maximal normal subgroup of | #72239415, #72239420, #72239426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55644);