Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437969 |
| 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 | C22×C4 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,12,26,11,16,10,15)(7,17,25,13,20,22,19,8)(9,14,27,36,23,29,24,30)(18,31,35,28,32,34,33,21);(1,2)(3,4)(5,7)(6,13)(8,16)(9,18)(10,25)(11,20)(12,19)(14,28)(15,22)(17,26)(21,29)(23,32)(24,35)(27,33)(30,34)(31,36);(2,4)(5,8,11,13)(6,7,16,20)(9,21,23,28)(10,17,12,22)(14,18,29,32)(15,25,26,19)(24,31,27,34)(30,35,36,33);(2,4)(5,9)(6,14)(7,18)(8,21)(10,24)(11,23)(12,27)(13,28)(15,30)(16,29)(17,31)(19,33)(20,32)(22,34)(25,35)(26,36);(9,23)(14,29)(18,32)(21,28)(24,27)(30,36)(31,34)(33,35)Polycyclic presentation:
2^9|a^2=fi,c^2=i,f^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312395, #72312417, #72312662, … |
| maximal normal subgroup of | #72312395, #72312417, #72312662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345168);