Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| 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 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,14)(7,11,17,24)(8,18)(9,13,21,26)(12,20)(15,22)(16,23,29,34)(19,31)(25,32)(27,33)(28,30)(35,36);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,22)(13,23)(14,24)(18,30)(19,27)(20,28)(21,29)(25,33)(26,34)(31,36)(32,35);(2,4)(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,25)(14,18)(16,27)(17,28)(21,32)(23,33)(24,30)(26,31)(29,35)(34,36);(2,4)(5,9)(6,13)(7,16)(8,19)(10,21)(11,23)(12,25)(14,26)(15,27)(17,29)(18,31)(20,32)(22,33)(24,34)(28,35)(30,36);(9,21)(13,26)(16,29)(19,32)(23,34)(25,31)(27,35)(33,36)Polycyclic presentation:
2^9|a^2=fi,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72267071, #72267077, #72280733, … |
| maximal normal subgroup of | #72267071, #72267077, #72280733, … |
| maximal quotient of | #72267143 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345196);