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 | |
8308 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)(13,14)(15,20)(16,27)(17,21)(18,23)(19,24)(22,26)(25,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,20)(16,23)(17,25)(18,27)(19,26)(21,28)(22,24);(6,9)(10,11)(13,16,19,17)(14,21,24,27)(15,25,26,23)(18,22,28,20);(5,6)(7,10)(8,9)(11,12)(13,17,19,16)(14,21,24,27)(15,25,26,23)(18,20,28,22);(5,6)(7,10)(8,9)(11,12)(13,15)(14,22)(16,23)(17,25)(18,21)(19,26)(20,24)(27,28);(5,7)(6,10)(8,12)(9,11)(13,18)(14,23)(15,21)(16,22)(17,20)(19,28)(24,25)(26,27)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304538, #72304554, #72304590, … |
| maximal normal subgroup of | #72304538, #72304554, #72304590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7640645);