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 | |
5568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(9,10,13,14)(11,12,15,16)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28);(2,5)(6,8)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72210404 |
| maximal normal subgroup of | #72210404 |
| maximal quotient of | #72210242, #72266737, #72266759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032904);