Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| 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 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3144 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)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22)(23,31)(24,27)(25,29)(26,30)(28,32)(33,36)(34,35);(2,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,23)(22,28)(24,32)(25,33)(26,34)(27,35)(29,31)(30,36);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,24,26,25)(22,29,30,27)(23,32,34,33)(28,31,36,35);(6,14)(11,20)(12,13)(18,19)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396872);