Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×2−1+4 | C23 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,23)(10,25)(11,16)(12,20)(13,22)(15,27)(18,29)(19,33)(21,28)(24,32)(26,31)(30,34)(35,36);(5,7,11,19)(6,12,16,29)(8,20,9,18)(10,26,27,21)(13,31,32,28)(14,33,23,17)(15,30,25,36)(22,34,24,35);(5,8,11,9)(6,13,16,32)(7,18,19,20)(10,24,27,22)(12,28,29,31)(14,15,23,25)(17,30,33,36)(21,35,26,34);(1,3,4,2)(7,19)(12,29)(17,33)(18,20)(21,26)(28,31)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72271508 |
| maximal normal subgroup of | #72271508 |
| maximal quotient of | #72271518 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402962);