Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3157 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)(13,16)(14,17)(15,18)(19,20)(21,22)(23,27)(24,26)(25,28)(29,32)(30,34)(31,33)(35,36);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(2,4)(7,13)(10,16)(14,19)(17,20)(21,23)(22,27)(24,31)(25,30)(26,33)(28,34)(29,35)(32,36);(6,11)(10,16)(12,18)(17,20)(23,30)(27,34)(31,36)(33,35);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(22,28)(26,29)(27,34)(33,35)Polycyclic presentation:
2^9|e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256919, #72256939, #72257082, … |
| maximal normal subgroup of | #72256919, #72256939, #72257082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343739);