Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(18,23)(19,26)(21,31)(24,28)(27,32)(29,30);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,18,22,23)(19,27,30,24)(20,31,25,21)(26,28,29,32);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,19,21,26,22,30,31,29)(18,24,20,32,23,27,25,28);(1,3,5,4)(2,8,6,7)(9,12,14,13)(10,15,11,16)(17,20,22,25)(18,21,23,31)(19,28,30,32)(24,29,27,26);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,15,11,16)(17,21,22,31)(18,25,23,20)(19,29,30,26)(24,32,27,28)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6959155);