Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,26)(17,27)(18,21)(19,22)(23,24)(25,28);(1,2)(3,4)(13,15,19,25)(14,20,22,28)(16,18,27,23)(17,24,26,21);(5,6,8,10)(7,9,12,11)(13,16,19,27)(14,17,22,26)(15,23,25,18)(20,21,28,24);(5,6)(7,9)(8,10)(11,12)(13,16)(14,17)(15,18)(19,27)(20,24)(21,28)(22,26)(23,25);(5,7)(6,9)(8,12)(10,11)(13,17)(14,16)(15,24)(18,20)(19,26)(21,25)(22,27)(23,28);(13,18)(14,21)(15,16)(17,28)(19,23)(20,26)(22,24)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72262811, #72284682, #72297209, … |
| maximal normal subgroup of | #72262811, #72284682, #72297209, … |
| maximal quotient of | #72262812, #72297220, #72297238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876749);