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 | #62842 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,15,10,17,9,16)(7,18,20,12,22,31,21,28)(8,14,23,29,25,13,24,30)(19,32,35,27,34,36,33,26);(1,2)(3,4)(5,7)(6,12)(8,19)(9,20)(10,22)(11,21)(13,26)(14,27)(15,18)(16,31)(17,28)(23,33)(24,35)(25,34)(29,32)(30,36);(2,4)(5,8)(6,13)(7,19)(9,23)(10,25)(11,24)(12,26)(14,17)(15,29)(16,30)(18,32)(20,33)(21,35)(22,34)(27,28)(31,36);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,23)(12,27)(13,17)(15,29)(16,30)(18,32)(20,35)(21,33)(22,34)(26,28)(31,36);(5,9,10,11)(6,15,17,16)(7,20,22,21)(8,23,25,24)(12,18,28,31)(13,29,14,30)(19,33,34,35)(26,32,27,36)Polycyclic presentation:
2^9|a^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,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 | #72318358, #72318377, #72318436, … |
| maximal normal subgroup of | #72318358, #72318377, #72318436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345174);