Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(6,7)(10,13)(14,16)(17,18,21,19)(20,22,24,23)(25,26,28,30)(27,31,32,29);(2,4)(6,7)(10,13)(11,15)(18,19)(20,24)(27,32)(29,31);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,22,24,23)(25,26)(27,29)(28,30)(31,32);(1,2)(3,4)(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^9|a^2=fg,c^2=fgh,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72328611, #72328744, #72328907, … |
| maximal normal subgroup of | #72328611, #72328744, #72328907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67306);