Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(17,18)(19,26)(20,22)(21,24)(23,30)(25,31)(27,32)(28,29);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(18,20)(19,27)(24,25)(29,30);(2,6)(3,8)(10,13)(14,16)(17,19,21,30,22,27,31,29)(18,23,25,26,20,28,24,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,28,27,23)(21,24,31,25)(26,30,32,29);(10,13)(14,16)(17,21,22,31)(18,24,20,25)(19,29,27,30)(23,26,28,32)Polycyclic presentation:
2^9|c^2=fgi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693543);