Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2482 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)(7,15,17,11)(8,19,20,12)(9,14)(10,13)(16,25)(18,28,31,24)(21,32,33,26)(22,27)(23,29)(30,35)(34,36);(2,4)(5,7,9,18,10,17,22,31)(6,11,14,24,13,15,27,28)(8,16,21,30,20,29,33,36)(12,25,26,35,19,23,32,34);(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,19)(14,26)(15,25)(17,29)(18,30)(22,33)(24,34)(27,32)(28,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,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 | #72251187, #72251198, #72251209, … |
| maximal normal subgroup of | #72251187, #72251198, #72251209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30738);