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 | |
4783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(14,16)(17,18)(19,22)(20,23)(21,28)(24,26)(25,29)(27,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,22)(21,27)(23,26)(25,30)(28,31)(29,32);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(18,20)(19,26)(21,29)(30,31);(1,3)(2,6)(4,8)(5,7)(17,20,23,18)(19,22,26,24)(21,28,29,25)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(17,21,23,29)(18,25,20,28)(19,27,26,32)(22,31,24,30);(1,2)(3,6)(4,7)(5,8)(17,22)(18,19)(20,26)(21,30)(23,24)(25,32)(27,28)(29,31)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7874148);