Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2543 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)(7,17)(8,19)(9,23)(10,16)(11,15)(12,14)(13,26)(18,30)(20,28)(21,24)(22,27)(25,31)(29,32)(33,35)(34,36);(1,2)(3,4)(5,7)(6,8)(9,24)(10,27)(11,19)(12,20)(13,30)(14,22)(15,28)(16,17)(18,25)(21,31)(23,36)(26,33)(29,34)(32,35);(2,4)(5,8)(6,7)(9,21)(10,17)(11,28)(12,22)(13,18)(14,20)(15,19)(16,27)(23,34)(24,31)(25,30)(26,35)(29,36)(32,33);(5,9,10,25,12,26,11,29)(6,13,15,31,14,23,16,32)(7,18,19,24,20,34,27,33)(8,21,17,30,22,35,28,36);(6,14)(8,22)(13,23)(15,16)(17,28)(21,35)(30,36)(31,32)Polycyclic presentation:
2^9|d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409352);