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 | |
7237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,8,9)(7,10,12,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(6,9)(10,11)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,7,8,12)(6,10,9,11)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,f^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 | #72312570, #72312575, #72312736, … |
| maximal normal subgroup of | #72312570, #72312575, #72312736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7882219);