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 | |
4997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(18,23)(19,24)(25,29)(27,30);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32);(2,5)(6,8)(17,19,22,24)(18,20,23,26)(21,30,31,27)(25,32,29,28);(3,7)(6,8)(17,20,22,26)(18,24,23,19)(21,28,31,32)(25,30,29,27);(1,3)(2,6)(4,7)(5,8)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21,22,31)(18,25,23,29)(19,27,24,30)(20,28,26,32)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7727141);