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 | |
5432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,8)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32);(1,2)(3,7)(4,8)(5,6)(10,14)(11,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,15)(14,16)(19,26)(22,23)(27,32)(30,31);(3,8)(4,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32);(1,4)(2,3)(5,7)(6,8)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32);(1,2)(3,7)(4,8)(5,6)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)Polycyclic presentation:
2^9|a^2=gh,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72333792 |
| maximal normal subgroup of | #72333792 |
| maximal quotient of | #72333828, #72333838, #72333962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9073950);