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 | #52004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,2,5,3)(4,6,8,7)(10,13)(14,15)(18,19)(23,25)(24,26)(30,31);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(21,29)(24,26)(27,32)(30,31);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,5,2)(4,7,8,6)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,8)(2,6,3,7)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284415, #72285519, #72285530, … |
| maximal normal subgroup of | #72284415, #72285519, #72285530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8101094);