Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,15,11,17,26,13)(7,18,24,35,25,36,22,30)(8,16,19,14,27,34,10,28)(12,29,32,20,33,23,31,21);(1,2,3,4)(5,7,19,29,9,24,27,20,11,25,10,23,26,22,8,21)(6,12,16,30,13,31,28,36,17,33,34,35,15,32,14,18);(6,13)(7,20)(8,19)(9,26)(10,27)(12,30)(14,28)(15,17)(18,31)(21,25)(22,23)(24,29)(32,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239852, #72239857, #72239863, … |
| maximal normal subgroup of | #72239852, #72239857, #72239863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55614);