Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140781 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1014 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)(5,6,9,15)(7,12)(8,11,23,27)(10,24,22,16)(13,31,14,20)(17,32)(18,30)(19,34)(21,26)(25,28)(29,36)(33,35);(2,4)(6,11)(7,17)(8,20)(10,22)(12,28)(13,16)(14,24)(15,27)(18,35)(19,36)(21,25)(23,31)(26,29)(30,34);(5,7)(6,12)(8,21)(9,18)(10,25)(11,26)(13,32)(14,33)(15,30)(16,34)(17,20)(19,22)(23,36)(24,28)(27,29)(31,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=hi,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,f^b=fh,h^b=hi,d^c=dh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239804, #72239809, … |
| maximal normal subgroup of | #72239802, #72239804, #72239809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55123);