Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2041 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
504 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)(7,19,23,36)(8,25)(9,29)(10,17)(11,31)(12,18)(13,20,34,26)(14,21,33,30)(15,27)(16,28)(22,32,24,35);(5,7,9,21,10,22,28,20,12,23,27,30,11,24,8,26)(6,13,16,32,31,14,29,19,18,34,25,35,17,33,15,36);(5,7,8,20,11,24,27,21,12,23,28,26,10,22,9,30)(6,14,15,35,17,13,25,19,18,33,29,32,31,34,16,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250577, #72250590, #72250618, … |
| maximal normal subgroup of | #72250577, #72250590, #72250618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56186);