Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13545 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2101 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
438 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)(5,6,12,18,11,29,9,28,14,8,26,30,13,17,10,16)(7,19,24,27,32,36,22,34,25,35,31,15,23,33,21,20);(2,4)(5,7)(6,15,8,27)(9,31)(10,24)(11,23)(12,21)(13,32)(14,25)(16,33,28,36)(17,20,29,34)(18,35,30,19)(22,26);(2,4)(5,8,11,29,14,6,13,17)(7,20,32,15,25,34,23,27)(9,30,26,28,10,18,12,16)(19,31,33,21,35,24,36,22)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239848, #72240847, #72240849, … |
| maximal normal subgroup of | #72239848, #72240847, #72240849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56560);