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 | |
7775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(14,18)(19,21)(20,26)(27,28);(7,11)(10,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,28)(22,25)(26,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26)Polycyclic presentation:
2^9|e^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930, #72331611, #72336917 |
| maximal normal subgroup of | #72326930, #72331611, #72336917 |
| maximal quotient of | #72326993, #72328209, #72329506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7555951);