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 | |
8204 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)(5,6,9,11)(7,8,12,10)(17,25)(21,26)(22,27)(24,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,26)(27,28);(2,4)(5,8,9,10)(6,7,11,12)(14,19)(16,23)(21,26)(24,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,18,20)(14,16,19,23)(17,22,25,27)(21,24,26,28);(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,26)(25,28);(13,17,18,25)(14,21,19,26)(15,22,20,27)(16,24,23,28)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7588556);