Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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×#3345 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,29)(26,32)(30,34)(31,35)(33,36);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,25,30)(22,27,29,34)(24,31,32,36)(26,33,28,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,28)(23,30)(24,32)(26,29)(27,33)(34,35);(6,12)(10,13)(11,20)(18,19)(22,29)(23,30)(26,28)(31,36)Polycyclic presentation:
2^9|c^2=hi,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280939, #72284364, #72284852, … |
| maximal normal subgroup of | #72280939, #72284364, #72284852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031563);