Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,25)(18,30)(19,34)(21,27)(23,31)(24,32)(26,28)(29,35)(33,36);(1,3,4,2)(6,12)(7,18)(9,10)(14,28)(15,16)(17,27)(19,33)(20,21)(22,32)(23,35)(24,25)(31,36);(2,3)(5,7)(6,13)(8,23)(9,20)(10,18)(11,21)(12,27)(14,31)(15,17)(16,30)(19,24)(22,34)(25,35)(26,33)(28,29)(32,36);(5,8,11,26)(6,14,16,32)(7,19,21,35)(9,25,10,24)(12,28,15,22)(13,29,30,34)(17,31,27,36)(18,33,20,23)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363947);