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 | |
7873 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)(5,6,9,11)(7,12,8,10)(13,14,18,21)(15,17,23,19)(16,20,24,27)(22,25,28,26);(1,2)(3,4)(7,8)(10,12)(16,24)(20,27)(22,28)(25,26);(5,7,9,8)(6,10,11,12)(13,15,18,23)(14,19,21,17)(16,22,24,28)(20,26,27,25);(5,6)(7,12)(8,10)(9,11)(13,15)(14,17)(16,22)(18,23)(19,21)(20,25)(24,28)(26,27);(5,8,9,7)(6,10,11,12)(13,16,18,24)(14,20,21,27)(15,22,23,28)(17,25,19,26);(5,6)(7,10)(8,12)(9,11)(13,17)(14,15)(16,25)(18,19)(20,22)(21,23)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72326380, #72326930, #72331611 |
| maximal normal subgroup of | #72326380, #72326930, #72331611 |
| maximal quotient of | #72326407, #72326443, #72326949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7945089);