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 | |
4935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,11)(12,15,16,14)(17,18,23,29)(19,27,21,24)(20,26,31,25)(22,28,32,30);(1,3,5,8)(2,6,4,7)(17,19)(18,24)(20,28)(21,23)(22,26)(25,32)(27,29)(30,31);(10,11)(14,15)(17,20,23,31)(18,25,29,26)(19,30,21,28)(22,27,32,24);(1,2,5,4)(3,6,8,7)(9,11,13,10)(12,15,16,14)(17,20)(18,26)(19,28)(21,30)(22,24)(23,31)(25,29)(27,32);(9,12,13,16)(10,14,11,15)(17,21,23,19)(18,27,29,24)(20,28,31,30)(22,25,32,26);(1,4,5,2)(3,7,8,6)(17,22)(18,28)(19,26)(20,24)(21,25)(23,32)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7711604);