Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(26,32)(33,36);(1,3,4,2)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,22)(23,26)(24,27)(25,28)(29,33)(30,32)(31,34)(35,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(21,23)(22,26)(24,29)(25,30)(27,33)(28,32)(31,35)(34,36);(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,33,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^b=fi,e^c=ei,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99501);