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 | |
5778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(2,5)(6,7)(10,15)(12,14)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,14)(10,11,15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72340652, #72340658, #72340660, … |
| maximal normal subgroup of | #72340652, #72340658, #72340660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618571);