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 | |
5464 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)(6,8)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30);(1,2,4,5)(3,6,7,8)(9,10,13,12)(11,15,16,14)(17,19,23,22)(18,20,24,26)(21,27,31,30)(25,28,29,32);(3,7)(6,8)(10,12)(14,15)(18,24)(19,22)(25,29)(27,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,26)(18,19,24,22)(21,28,31,32)(25,27,29,30);(17,21,23,31)(18,25,24,29)(19,27,22,30)(20,28,26,32);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,22,23,19)(18,26,24,20)(21,30,31,27)(25,32,29,28)Polycyclic presentation:
2^9|b^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284341, #72284474, #72284513, … |
| maximal normal subgroup of | #72284341, #72284474, #72284513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8102959);