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 | |
5423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,19,22,25)(18,20,23,28)(21,30,31,26)(24,32,29,27);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,25)(24,29)(26,30);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,30)(28,32)Polycyclic presentation:
2^9|b^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72330757 |
| maximal normal subgroup of | #72330757 |
| maximal quotient of | #72330870, #72331850, #72332021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227765);