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 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,14,11,17,25,35,12,18,10,15,26,29,9,16)(7,19,21,30,23,36,20,13,24,32,22,33,28,31,27,34);(1,2)(3,4)(5,7,11,28,12,24,26,23)(6,13,29,33,18,34,17,30)(8,20,25,21,10,27,9,22)(14,31,16,19,15,36,35,32);(2,4)(6,14,18,15)(7,20,24,27)(8,25)(9,10)(11,26)(13,30)(16,29,35,17)(21,28,22,23)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250226, #72250230, #72250256, … |
| maximal normal subgroup of | #72250226, #72250230, #72250256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56591);