Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246239 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1600 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1389 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)(7,16)(8,13)(9,12)(10,19)(11,15)(14,17)(18,20)(24,31)(26,32)(27,34)(33,36);(1,2)(3,4)(5,7,11,17)(6,12,15,20)(8,9,10,18)(13,14,19,16)(21,22,25,28)(23,26,30,32)(24,27,31,34)(29,33,35,36);(2,4)(5,8,11,10)(6,13,15,19)(7,9,17,18)(12,14,20,16)(21,23,25,30)(22,26,28,32)(24,29,31,35)(27,33,34,36);(5,9,11,18)(6,14,15,16)(7,8,17,10)(12,13,20,19)(21,24)(22,27)(23,29)(25,31)(26,33)(28,34)(30,35)(32,36)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^c=ei,e^d=eh,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153502);