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 | |
5453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,14,13)(11,12,16,15)(17,18)(19,21)(20,24)(22,25)(23,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,8)(5,7)(10,13)(11,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11,14,16)(10,15,13,12)(20,28)(24,31)(26,32)(29,30);(3,4)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,31)(27,32);(1,4,5,3)(2,6,7,8)(9,13,14,10)(11,12,16,15)(17,21,22,23)(18,19,25,27)(20,29,28,30)(24,32,31,26);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=g,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 | #72265951, #72265955, #72265962, … |
| maximal normal subgroup of | #72265951, #72265955, #72265962, … |
| maximal quotient of | #72266502, #72266567, #72333451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9073965);