Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500268 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61675 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,10)(5,8)(6,20)(7,12)(9,16)(11,21)(14,29)(15,25)(17,22)(18,23)(19,27)(24,32)(26,31)(28,30);(2,8)(3,14)(4,5)(6,21)(9,24)(10,12)(11,27)(13,28)(15,32)(16,31)(17,30)(20,23)(22,26)(25,29);(1,3,5,15,7,16,4,17)(2,9,10,25,12,13,8,22)(6,14,19,30,21,31,18,32)(11,26,27,28,20,29,23,24);(1,4)(3,15)(5,7)(6,18)(8,10)(9,13)(14,30)(16,17)(19,21)(23,27)(26,29)(31,32)Polycyclic presentation:
2^9|c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407464);