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 | |
6288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(10,13)(14,16)(17,18)(19,25)(20,29)(21,30)(22,23)(24,28)(26,27)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(20,30)(24,25)(26,32);(2,5)(6,8)(11,15)(14,16)(17,19,22,28)(18,24,23,25)(20,26,30,32)(21,27,29,31);(9,10)(11,14)(12,13)(15,16)(17,20)(18,21)(19,26)(22,30)(23,29)(24,27)(25,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,26)(21,27)(22,28)(23,25)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,20)(19,27)(22,29)(23,30)(24,26)(25,32)(28,31)Polycyclic presentation:
2^9|c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256737, #72261883, #72262049, … |
| maximal normal subgroup of | #72256737, #72261883, #72262049, … |
| maximal quotient of | #72256793, #72263415 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022944);