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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,16)(7,17,21,28)(8,22,25,13)(9,15,10,14)(12,19,29,20)(18,30,34,35)(23,27,33,36)(24,31,26,32);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,24)(22,35)(25,33)(26,34)(27,31)(32,36);(2,4)(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,27)(14,17)(15,28)(16,29)(22,36)(23,24)(25,34)(26,33)(30,31)(32,35);(5,8,9,24,11,25,10,26)(6,13,14,31,16,22,15,32)(7,18,20,23,21,34,19,33)(12,27,17,30,29,36,28,35);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,25,26)(12,28,29,17)(13,31,22,32)(18,33,34,23)(27,35,36,30)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=i,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, 7409356);