Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,14,12,7,11,16)(8,19,21,31,24,34,23,35)(10,26,25,18,29,32,28,15)(13,27,30,22,17,33,20,36);(1,2)(3,4)(5,7)(6,12)(8,20)(9,14)(10,15)(11,16)(13,23)(17,21)(18,29)(19,33)(22,31)(24,30)(25,32)(26,28)(27,34)(35,36);(1,3,4,2)(5,8)(6,13,7,17)(9,23)(10,27)(11,21)(12,24)(14,20,16,30)(15,31,18,35)(19,32,34,26)(22,28)(25,36)(29,33)Polycyclic presentation:
2^9|a^2=g,c^2=dh,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321483, #72321547, #72321571, … |
| maximal normal subgroup of | #72321483, #72321547, #72321571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55854);