Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,15,25,29,9,16,11,14,26,30,10,17,23,12)(7,18,19,13,27,32,24,35,22,34,36,31,21,28,20,33);(1,2,3,4)(6,12)(8,23)(9,26)(10,25)(13,28)(14,16)(15,17)(18,33)(19,20)(21,27)(24,36)(29,30)(31,32)(34,35);(5,7)(6,13)(8,24)(9,19)(10,27)(11,22)(12,28)(14,31)(15,34)(16,32)(17,35)(18,30)(20,26)(21,25)(23,36)(29,33);(6,14)(12,16)(13,31)(15,30)(17,29)(18,34)(28,32)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250557, #72250645, #72250656, … |
| maximal normal subgroup of | #72250557, #72250645, #72250656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401723);