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 | |
4951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,13,15)(11,12,16,14)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,16)(14,15);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32);(1,4,5,2)(3,8,7,6)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^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, 8380144);