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 | |
6855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6,9,10)(7,11,12,8)(13,14)(15,16)(17,21)(18,20)(19,22)(23,24)(25,27)(26,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,19)(16,20)(17,23)(18,22)(21,26)(24,27)(25,28);(5,6,9,10)(7,11,12,8)(14,20)(15,22)(17,25)(24,26);(7,12)(8,11)(13,16,18,19)(14,15,20,22)(17,26,25,24)(21,28,27,23);(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,20)(17,23)(18,22)(21,26)(24,27)(25,28);(5,8)(6,7)(9,11)(10,12)(13,17,18,25)(14,21,20,27)(15,23,22,28)(16,24,19,26)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=h,f^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 | #72312780, #72312785, #72312916, … |
| maximal normal subgroup of | #72312780, #72312785, #72312916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982754);