Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3060 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)(21,22)(23,27)(24,26)(25,28)(29,32)(30,34)(31,33)(35,36);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,27)(24,31)(25,30)(26,33)(28,34)(29,35)(32,36);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(22,28)(23,30)(26,29)(31,36);(8,16)(11,18)(14,20)(17,19)(23,30)(27,34)(31,36)(33,35)Polycyclic presentation:
2^9|b^2=h,c^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72307791, #72307881, #72308427 |
| maximal normal subgroup of | #72307791, #72307881, #72308427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6478771);