Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,19,29)(8,20)(9,13)(10,14)(11,18,26,23)(12,21)(16,24)(17,30,32,36)(22,34)(25,33,27,35)(28,31);(2,4)(5,7,16,18,10,19,9,23)(6,11,24,15,14,26,13,29)(8,17,31,33,21,32,22,35)(12,27,34,36,20,25,28,30);(2,4)(5,8,16,31,10,21,9,22)(6,12,24,34,14,20,13,28)(7,17,18,33,19,32,23,35)(11,25,15,30,26,27,29,36);(8,21)(12,20)(17,32)(22,31)(25,27)(28,34)(30,36)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250584, #72250593, #72250605, … |
| maximal normal subgroup of | #72250584, #72250593, #72250605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265812);