Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1136 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)(7,18)(8,23)(9,13)(10,28)(11,14)(12,19)(15,24)(16,26)(17,25)(20,35)(21,30)(22,34)(27,31)(29,32)(33,36);(1,3,4,2)(5,7)(6,12)(8,20)(9,27)(10,29)(11,22)(13,30)(14,31)(15,32)(16,34)(17,35)(18,23)(19,24)(21,25)(26,36)(28,33);(6,13)(8,24)(9,26)(10,25)(12,30)(14,15)(16,28)(17,23)(18,35)(19,20)(21,29)(27,36)(31,32)(33,34);(6,14)(12,31)(13,15)(16,23)(17,28)(18,34)(30,32)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250446, #72250460, #72250487, … |
| maximal normal subgroup of | #72250446, #72250460, #72250487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401496);