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 | |
2591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,12,16)(7,19,23,34)(8,24,15,9)(10,18,11,17)(13,28,33,22)(14,29,30,26)(20,31,25,21)(27,32,35,36);(1,2)(3,4)(5,7)(6,13)(8,25)(9,27)(10,28)(11,22)(12,23)(14,21)(15,20)(16,33)(17,34)(18,19)(24,35)(26,36)(29,32)(30,31);(2,4)(5,8)(6,14)(7,20)(9,18)(10,26)(11,29)(12,15)(13,31)(16,30)(17,24)(19,35)(21,33)(22,36)(23,25)(27,34)(28,32);(5,9,10,14,12,24,11,30)(6,15,17,29,16,8,18,26)(7,21,22,27,23,31,28,35)(13,32,19,20,33,36,34,25);(6,16)(8,15)(13,33)(17,18)(19,34)(20,25)(26,29)(32,36)Polycyclic presentation:
2^9|a^2=i,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, 7409353);