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 | |
8764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6,10,12)(7,9,8,11)(16,23)(20,25)(21,27)(26,28);(1,2)(3,4)(5,7,10,8)(6,11,12,9)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28);(6,12)(7,8)(14,18)(19,24)(20,25)(26,28);(5,6,10,12)(7,11,8,9)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28);(5,8,10,7)(6,9,12,11)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28);(5,9)(6,7)(8,12)(10,11)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304566, #72304570, #72304580, … |
| maximal normal subgroup of | #72304566, #72304570, #72304580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617220);