Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2088 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,18,9,27,8,16,13,19,26,14,12,17,10,29)(7,20,24,31,28,34,22,30,25,33,35,36,23,15,21,32);(1,2,4,3)(6,14,19,18)(7,21,25,22)(8,11)(9,12)(10,26)(16,17,29,27)(20,33)(23,24,28,35)(30,36)(31,32);(1,3,4,2)(5,7,8,22,12,23,11,24,13,25,10,21,9,28,26,35)(6,15,16,31,17,33,18,32,19,34,29,36,27,20,14,30)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=i,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250768, #72250775, #72250785, … |
| maximal normal subgroup of | #72250768, #72250775, #72250785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56278);