Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,19,20,18)(21,22,25,29,26,30,23,27)(24,32,31,35,33,36,34,28);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,25)(22,27,30,29)(24,31,33,34)(28,32,35,36);(2,4)(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20)(21,24,26,33)(22,28,30,35)(23,31,25,34)(27,32,29,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174904);