Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| 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:
(5,6,11,16)(7,18)(8,9,24,10)(12,28,19,17)(13,30)(14,27,15,23)(20,33)(21,34)(22,31)(25,36)(26,32)(29,35);(1,2)(3,4)(5,7,19,21,11,22,12,29)(6,13,17,36,16,33,28,32)(8,18,14,34,24,31,15,35)(9,25,27,20,10,26,23,30);(2,4)(5,8,23,17,19,15,9,6,11,24,27,28,12,14,10,16)(7,13,26,35,29,36,20,18,22,33,25,34,21,32,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250823, #72250834, #72250838, … |
| maximal normal subgroup of | #72250823, #72250834, #72250838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56370);