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 | |
424 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)(7,18,22,31)(8,12)(9,14)(10,27)(11,17)(13,21,32,19)(15,26)(16,25)(20,33,36,29)(23,30,35,34)(24,28);(1,2)(3,4)(6,12,17,15)(7,19,22,21)(8,24)(9,26)(10,25)(13,29)(14,27,28,16)(20,35,36,23)(30,34)(32,33);(2,4)(5,7,9,19,10,23,26,36,11,22,24,21,25,35,8,20)(6,13,12,18,16,33,28,34,17,32,15,31,27,29,14,30)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,h^b=hi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239852, #72240846, #72240849, … |
| maximal normal subgroup of | #72239852, #72240846, #72240849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56362);