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 | |
5391 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)(12,13)(14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32);(1,2)(3,4)(5,7)(6,8)(10,13)(14,15)(18,22)(23,25)(24,28)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31);(3,8)(4,6)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30);(1,4)(2,3)(5,6)(7,8)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31);(1,2)(3,4)(5,7)(6,8)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)Polycyclic presentation:
2^9|c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72334246 |
| maximal normal subgroup of | #72334246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8875732);