Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,33)(24,29)(25,30)(26,31)(27,32)(28,35)(34,36);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,27)(25,35)(26,34)(29,30)(31,33)(32,36);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,23)(22,29,32,33)(25,35,26,34)(28,31,36,30);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,32,31)(23,34,24,35)(28,33,36,29);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,31,32,30)(23,35,24,34)(28,33,36,29)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384544);