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 | |
5854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,32)(29,31);(1,3,5,8)(2,6,7,4)(9,11,13,14)(10,12,15,16)(17,19,23,26)(18,24,22,21)(20,29,30,27)(25,32,28,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(18,22)(19,26)(20,30)(31,32);(2,7)(3,8)(11,14)(12,16)(17,20)(18,25)(19,27)(21,32)(22,28)(23,30)(24,31)(26,29);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,21)(18,19)(20,31)(22,26)(23,24)(25,29)(27,28)(30,32);(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,22,23,18)(19,21,26,24)(20,28,30,25)(27,32,29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,f^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,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312186, #72319710, #72319805, … |
| maximal normal subgroup of | #72312186, #72319710, #72319805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841808);