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 |
| prime radical | #62988 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,16)(7,12,20,27)(8,13,23,30)(9,15,10,14)(17,31,34,24)(18,26,19,25)(21,29,22,28)(32,35,33,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,24)(14,25)(15,26)(16,27)(21,32)(22,33)(23,34)(28,35)(29,36)(30,31);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,24)(14,28)(15,29)(16,30)(18,32)(19,33)(20,34)(25,35)(26,36)(27,31);(1,2)(3,4)(5,9,11,10)(6,14,16,15)(7,18,20,19)(8,21,23,22)(12,25,27,26)(13,28,30,29)(17,32,34,33)(24,35,31,36);(2,4)(5,10,11,9)(6,15,16,14)(7,19,20,18)(8,22,23,21)(12,26,27,25)(13,29,30,28)(17,33,34,32)(24,36,31,35)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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 | #72306238, #72306248, #72306273, … |
| maximal normal subgroup of | #72306238, #72306248, #72306273, … |
| maximal quotient of | #72306466 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480924);