Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,11,15,12,7,10,16)(8,19,22,17,23,30,21,13)(9,14,26,29,27,18,25,24)(20,32,35,31,36,33,34,28);(1,3,4,2)(5,7)(6,12)(8,17)(9,24)(10,16)(11,15)(13,23)(14,25)(18,26)(19,22)(20,33)(21,30)(27,29)(28,34)(31,35)(32,36);(5,8)(6,13)(7,17)(9,20)(10,21)(11,22)(12,23)(14,28)(15,19)(16,30)(18,31)(24,33)(25,34)(26,35)(27,36)(29,32);(5,9,10,25,12,27,11,26)(6,14,16,24,7,18,15,29)(8,20,21,34,23,36,22,35)(13,28,30,33,17,31,19,32);(5,10,12,11)(6,15,7,16)(8,21,23,22)(9,25,27,26)(13,19,17,30)(14,29,18,24)(20,34,36,35)(28,32,31,33)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279230, #72286027, #72286096, … |
| maximal normal subgroup of | #72279230, #72286027, #72286096, … |
| maximal quotient of | #72279507, #72309525, #72322213 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289206);