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 | |
7300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(14,19)(15,23)(21,22)(25,26);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,24,23,20)(16,22,17,21)(25,27,26,28);(6,8)(10,11)(13,15,18,23)(14,20,19,24)(16,26,17,25)(21,28,22,27);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,21,19,22)(15,25,23,26)(20,27,24,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,24)(25,27)(26,28);(5,8,9,6)(7,11,12,10)(13,17,18,16)(14,22,19,21)(15,26,23,25)(20,28,24,27)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713786);