Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
887 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)(6,11)(7,16)(8,21)(9,25)(13,22)(14,15)(17,28)(18,31)(19,20)(26,30)(32,35)(33,34);(5,6,10,15)(7,17)(8,22,21,26)(9,14,25,11)(12,19)(13,24,30,23)(16,29)(18,27)(20,28)(31,32)(33,35)(34,36);(1,3,4,2)(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,27)(14,28)(15,29)(16,25)(21,34)(22,35)(23,31)(24,33)(26,32)(30,36);(5,8,10,21)(6,13,15,30)(7,18,20,34)(9,24,25,23)(11,26,14,22)(12,27,29,36)(16,31,19,33)(17,32,28,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295956);