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 | #2563 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,23)(9,25,28,32)(10,17)(11,16)(12,18)(13,21)(14,22)(15,27,31,26)(20,29,35,36)(24,33,34,30);(1,2)(3,4)(5,7,10,21,12,22,11,8)(6,13,16,19,18,23,17,14)(9,24,26,35,28,34,27,20)(15,30,25,36,31,33,32,29);(2,4)(5,8,11,22,12,21,10,7)(6,14,17,23,18,19,16,13)(9,24,26,35,28,34,27,20)(15,30,25,36,31,33,32,29);(5,9)(6,15)(7,20)(8,24)(10,27)(11,26)(12,28)(13,29)(14,30)(16,32)(17,25)(18,31)(19,33)(21,34)(22,35)(23,36)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332163);