Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508561 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29942 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,7,9,5,11)(3,14,16,24,18,29,22,8)(4,10,20,26,21,12,19,27)(15,25,31,28,32,30,17,23);(1,3)(2,8,9,24)(4,15)(5,16)(6,22)(7,18)(10,23,12,28)(11,14,13,29)(17,20)(19,31)(21,32)(25,26,30,27);(2,9)(3,15,18,32)(4,19)(5,6)(8,23,24,28)(10,26)(12,27)(14,30,29,25)(16,17,22,31)(20,21);(1,4,5,19,7,21,6,20)(2,10,11,27,9,12,13,26)(3,15,16,31,18,32,22,17)(8,23,14,25,24,28,29,30)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=eh,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=de,e^c=ei,g^c=ghi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415757);