Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500256 |
| 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 | #61644 | 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 | |
867 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)(6,11)(7,12)(9,16)(14,27)(15,30)(17,29)(18,26)(19,23)(20,24)(21,28)(22,31)(25,32);(2,8)(3,14)(4,5)(9,22)(10,12)(11,23)(13,29)(18,19)(24,25)(26,28)(27,30)(31,32);(1,3,5,15,7,14,4,17)(2,9,10,25,12,24,8,22)(6,20,19,32,21,16,18,31)(11,27,26,30,28,13,23,29);(1,4)(3,15)(5,7)(6,18)(8,10)(9,24)(13,27)(14,17)(16,31)(19,21)(20,32)(23,26)Polycyclic presentation:
2^9|c^2=e,e^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,d^c=dh,f^c=fi,g^c=gi,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, 407452);