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 | |
5096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,12,15,16)(17,18,24,27)(19,20,28,30)(21,25,29,32)(22,23,26,31);(2,6)(3,8)(10,14)(11,15)(17,19,24,28)(18,20,27,30)(21,31,29,23)(22,32,26,25);(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,26)(22,29)(23,25)(24,30)(27,28)(31,32);(2,6)(4,7)(9,12)(10,11)(13,16)(14,15)(17,21,24,29)(18,25,27,32)(19,23,28,31)(20,26,30,22);(1,3)(2,4)(5,8)(6,7)(17,22,24,26)(18,23,27,31)(19,25,28,32)(20,29,30,21);(1,4)(2,3)(5,7)(6,8)(17,23)(18,26)(19,21)(20,25)(22,27)(24,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8380081);