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 | |
5433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,13)(14,15)(17,18,21,25)(19,24,26,22)(20,28,30,23)(27,31,29,32);(3,7)(5,8)(9,11,14,13)(10,12,15,16)(19,26)(22,24)(27,29)(31,32);(1,2)(3,5)(4,6)(7,8)(17,19)(18,22)(20,27)(21,26)(23,31)(24,25)(28,32)(29,30);(9,12)(10,13)(11,15)(14,16)(17,20,21,30)(18,23,25,28)(19,27,26,29)(22,31,24,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,24)(20,29)(21,26)(22,25)(23,31)(27,30)(28,32);(1,2)(3,5)(4,6)(7,8)(9,13,14,11)(10,12,15,16)(17,18)(19,22)(20,28)(21,25)(23,30)(24,26)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72263176, #72333435 |
| maximal normal subgroup of | #72263176, #72333435 |
| maximal quotient of | #72263226, #72333470, #72333482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9143700);