Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,20,26)(8,13,23,30)(9,15,10,14)(12,21,27,19)(18,32,34,31)(22,28,24,29)(25,35,36,33);(1,2)(3,4)(5,7)(6,12)(8,18)(9,21)(10,19)(11,20)(13,25)(14,26)(15,17)(16,27)(22,35)(23,34)(24,33)(28,32)(29,31)(30,36);(2,4)(5,8)(6,13)(7,18)(9,24)(10,22)(11,23)(12,25)(14,28)(15,29)(16,30)(17,31)(19,35)(20,34)(21,33)(26,32)(27,36);(6,14)(9,10)(12,26)(13,28)(15,16)(17,27)(19,21)(22,24)(25,32)(29,30)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251058, #72251074, #72251087, … |
| maximal normal subgroup of | #72251058, #72251074, #72251087, … |
| maximal quotient of | #72314500 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301119);