Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3115 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,26,29)(23,28,31,36)(24,30,25,27)(32,34,33,35);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(22,27)(23,31)(24,25)(28,35)(29,30)(34,36);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,23)(22,28)(24,33)(25,32)(26,31)(27,34)(29,36)(30,35);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(22,29)(27,30)(28,36)(34,35);(7,13)(10,16)(14,19)(17,20)(21,24,26,25)(22,30,29,27)(23,32,31,33)(28,34,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433970);