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 |
| prime radical | #40512 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,12)(11,15,16,14)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32);(1,2)(3,7)(4,8)(5,6)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,22)(21,28)(24,26)(25,32)(27,30)(29,31);(2,6)(4,7)(18,24)(19,20)(25,29)(27,28);(1,3)(2,4)(5,8)(6,7)(9,12,13,10)(11,15,16,14)(17,20,22,19)(18,23,24,26)(21,28,30,27)(25,31,29,32);(1,3)(2,4)(5,8)(6,7)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32);(1,4)(2,3)(5,7)(6,8)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,31)(29,30)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7712034);