Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
978 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)(5,6,11,18,13,17,12,34)(7,19,23,24,21,29,35,36)(8,22,25,33,26,15,14,28)(9,27,30,20,10,32,31,16);(5,7)(6,14)(8,18)(9,28)(10,33)(11,23)(12,35)(13,21)(15,31)(16,36)(17,25)(19,27)(20,24)(22,30)(26,34)(29,32);(2,4)(5,8)(6,15)(7,20)(9,29)(10,19)(11,14)(12,25)(13,26)(16,21)(17,22)(18,33)(23,27)(24,30)(28,34)(31,36)(32,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dhi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224865, #72224936, #72226819, … |
| maximal normal subgroup of | #72224865, #72224936, #72226819, … |
| maximal quotient of | #72224863, #72237264, #72280263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55141);