Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,17,12,19,27,18,13,20,9,28,8,16,10,30)(7,21,25,32,22,33,29,34,26,36,23,15,31,35,24,14);(1,2,3,4)(5,7)(6,14,20,34)(8,22)(9,24)(10,23)(11,29)(12,31)(13,26)(15,16,32,19)(17,33,28,35)(18,21,30,36)(25,27);(5,7)(6,15)(8,22)(9,29)(10,25)(11,24)(12,31)(13,26)(14,16)(17,36)(18,33)(19,34)(20,32)(21,28)(23,27)(30,35);(5,8,13,12)(6,16,20,19)(7,22,26,31)(9,27,11,10)(14,32,34,15)(17,30,28,18)(21,33,36,35)(23,24,25,29)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264744, #72264764, #72264781, … |
| maximal normal subgroup of | #72264744, #72264764, #72264781, … |
| maximal quotient of | #72285533, #72285656, C2×#494559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401755);