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 | |
5696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,12,13)(19,25)(23,26)(27,31)(30,32);(2,3)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30);(2,3)(6,7)(10,16)(11,12)(18,20)(23,26)(24,28)(30,32);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,13,16,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(1,3,5,2)(4,6,8,7)(9,12,14,11)(10,13,16,15)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,16)(14,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72265951, #72265957, #72265960, … |
| maximal normal subgroup of | #72265951, #72265957, #72265960, … |
| maximal quotient of | #72266527, #72266571 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976133);