Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,12)(8,13)(9,23)(10,14)(11,15)(16,29)(17,33)(18,25)(19,26)(20,34)(21,27)(22,28)(24,32)(30,35)(31,36);(1,2)(3,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,24)(14,25)(15,26)(20,30)(21,31)(22,32)(23,33)(27,35)(28,29)(34,36);(2,4)(5,8)(6,13)(7,16)(9,20)(10,21)(11,22)(12,24)(14,27)(15,28)(17,30)(18,31)(19,32)(23,34)(25,35)(26,29)(33,36);(1,2)(3,4)(5,9,11,10)(6,14,15,23)(7,17,19,18)(8,20,22,21)(12,25,26,33)(13,27,28,34)(16,30,32,31)(24,35,29,36);(2,4)(5,10,11,9)(6,14,15,23)(7,18,19,17)(8,21,22,20)(12,25,26,33)(13,27,28,34)(16,31,32,30)(24,35,29,36)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72267070, #72267071, #72267082, … |
| maximal normal subgroup of | #72267070, #72267071, #72267082, … |
| maximal quotient of | #72267130, #72267147, #72306408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480914);