Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,9)(8,22)(10,18)(11,16)(12,17)(13,15)(14,26)(19,30)(20,29)(21,28)(23,35)(24,34)(25,33)(27,32)(31,36);(5,7,12,20)(6,13,17,28)(8,23)(9,16,29,18)(10,21,11,15)(14,30)(19,27)(22,24)(25,31)(26,34)(32,36)(33,35);(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,27)(13,30)(15,31)(16,32)(17,33)(18,22)(20,23)(21,34)(28,35)(29,36);(2,4)(5,9,10,28,12,29,11,13)(6,15,16,20,17,21,18,7)(8,24,25,35,27,36,26,30)(14,31,32,23,33,34,22,19)Polycyclic presentation:
2^9|b^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328644);