Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
596 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)(7,16)(8,13)(9,22)(10,20)(11,15)(12,19)(14,25)(17,31)(18,33)(21,23)(24,30)(26,36)(27,34)(28,35)(29,32);(1,3,4,2)(5,6)(7,12)(8,20)(9,23)(10,13)(11,15)(14,26)(16,32)(17,30)(18,29)(19,33)(21,22)(24,34)(25,36)(27,28)(31,35);(5,7,10,27,11,18,8,17)(6,12,13,28,15,29,20,30)(9,24,25,32,26,35,21,19)(14,31,22,33,23,34,36,16);(5,8,11,10)(6,13,15,20)(7,17,18,27)(9,21,26,25)(12,28,29,30)(14,22,23,36)(16,31,33,34)(19,35,32,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=eh,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250191, #72250204, #72250219, … |
| maximal normal subgroup of | #72250191, #72250204, #72250219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402003);