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 | |
7343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,18,16)(15,19,21,22)(17,20,25,24)(23,26,27,28);(1,2)(3,4)(5,7,11,9)(6,8,12,10)(15,21)(19,22)(23,27)(26,28);(5,8,11,10)(6,9,12,7)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(5,7,11,9)(6,8,12,10);(5,9,11,7)(6,8,12,10)(13,16,18,14)(15,22,21,19)(17,20,25,24)(23,26,27,28);(5,10,11,8)(6,9,12,7)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72280838, #72281016, #72315027, … |
| maximal normal subgroup of | #72280838, #72281016, #72315027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7795810);