Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,29)(24,32)(25,27)(26,31)(28,33)(30,35)(34,36);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,24,31,25,30,33,36)(22,26,28,29,27,34,32,35);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(22,27)(26,34)(28,32)(29,35);(6,11)(10,16)(12,18)(17,20);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,30)(26,34)(29,35)(31,36)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284663, #72285043, #72285246, … |
| maximal normal subgroup of | #72284663, #72285043, #72285246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344419);