Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500092 |
| 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×#3514 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,25,29,26,30,24,31)(23,32,33,27,35,36,34,28);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,27)(24,33)(25,34)(26,35)(28,30)(29,32)(31,36);(6,11)(7,13)(8,15)(16,20)(21,23)(22,28)(24,33)(25,34)(26,35)(27,30)(29,36)(31,32);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(21,24,26,25)(22,29,30,31)(23,33,35,34)(27,32,28,36);(7,13)(10,17)(14,19)(16,20)(22,30)(27,28)(29,31)(32,36)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 6431977);