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 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,7)(9,10,13,14)(11,12,15,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,11,14,15)(21,29)(23,30)(26,32)(28,31);(2,5)(6,8)(10,14)(11,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,27)(21,23)(22,24)(26,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,21)(18,23)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72241005, #72252235, #72282959, … |
| maximal normal subgroup of | #72241005, #72252235, #72282959, … |
| maximal quotient of | #72283067, #72333358, #72334056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840076);