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 | |
7716 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)(5,6,9,10)(7,11,12,8)(13,14,18,19)(15,17,21,26)(16,20,25,27)(22,24,23,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(5,6,9,10)(7,11,12,8)(14,19)(17,26)(20,27)(24,28);(7,12)(8,11)(13,15,18,21)(14,17,19,26)(16,23,25,22)(20,28,27,24);(5,7)(6,8)(9,12)(10,11)(13,16)(14,20)(15,22)(17,24)(18,25)(19,27)(21,23)(26,28);(5,8)(6,7)(9,11)(10,12)(13,17)(14,21)(15,19)(16,24)(18,26)(20,23)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72312603, #72312633, #72312736, … |
| maximal normal subgroup of | #72312603, #72312633, #72312736, … |
| maximal quotient of | #72336923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982502);