Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| 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×#3188 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,31)(25,33)(26,29)(28,30)(32,35)(34,36);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,26,25)(22,27,29,33)(24,30,32,34)(28,35,36,31);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33);(6,12)(10,13)(11,18)(17,19)(21,23,26,25)(22,27,29,33)(24,30,32,34)(28,35,36,31);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714573);