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 | |
6088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10,13,14)(11,15,16,12)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32);(3,7)(6,8)(10,14)(12,15)(18,23)(20,25)(24,26)(28,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(18,23)(19,27)(21,31)(28,30);(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,28)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,25)(18,21,23,31)(19,28,27,30)(24,32,26,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,25)(19,29)(20,23)(22,31)(24,28)(26,30)(27,32)Polycyclic presentation:
2^9|a^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72219287, #72249434, #72249526, … |
| maximal normal subgroup of | #72219217, #72219285, #72219287, … |
| maximal quotient of | #72219215, #72262663, #72262816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8822401);