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 | |
5912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,26)(21,30)(22,24)(25,28)(27,31)(29,32);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,15,12)(17,19,22,28)(18,23,24,25)(20,27,29,30)(21,26,31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(18,24)(20,29)(23,25)(27,30);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,25)(20,27)(21,26)(22,28)(23,24)(29,30)(31,32);(17,20)(18,26)(19,27)(21,25)(22,29)(23,31)(24,32)(28,30);(17,21)(18,27)(19,26)(20,25)(22,31)(23,29)(24,30)(28,32)Polycyclic presentation:
2^9|b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72262090, #72262098, #72262211, … |
| maximal normal subgroup of | #72262090, #72262098, #72262211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550779);