Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | |
2588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,12)(7,16)(8,20)(9,10)(14,21)(15,24)(17,31)(18,19)(25,27)(26,35)(29,30)(33,34);(1,2)(3,4)(5,6)(7,13)(8,21)(9,12)(10,24)(11,15)(14,23)(16,27)(17,32)(18,25)(19,28)(20,29)(22,30)(26,31)(33,35)(34,36);(2,4)(5,7)(6,13)(8,17)(9,18)(10,16)(11,19)(12,25)(14,26)(15,28)(20,34)(21,32)(22,33)(23,31)(24,27)(29,36)(30,35);(5,8,10,23,11,20,9,22)(6,14,12,21,15,30,24,29)(7,17,16,31,19,34,18,33)(13,26,25,32,28,35,27,36);(5,9,11,10)(6,12,15,24)(7,18,19,16)(8,22,20,23)(13,25,28,27)(14,21,30,29)(17,33,34,31)(26,32,35,36)Polycyclic presentation:
2^9|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,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, 7409348);