Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(1,3)(2,5)(4,8)(6,7)(17,18,21,22)(19,23,26,25)(20,24,29,28)(27,30,32,31);(2,6)(5,7)(9,10,12,14)(11,13,16,15)(18,22)(19,26)(24,28)(27,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|b^2=i,c^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249417, #72249540, #72249545, … |
| maximal normal subgroup of | #72249417, #72249540, #72249545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7620622);