Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500894 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24299 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3067 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13)(3,14,11,6)(4,18)(5,17,7,16)(9,27)(10,25,12,24)(15,19)(20,28)(21,30)(22,31)(23,32)(26,29);(1,3,6,2,8,11,14,13)(4,9,20,29,23,30,22,15)(5,16,24,10,7,17,25,12)(18,26,31,27,32,19,28,21);(1,4)(2,9)(3,15)(5,19)(6,20)(7,26)(8,23)(10,28)(11,29)(12,31)(13,30)(14,22)(16,32)(17,18)(21,24)(25,27);(1,5,6,24,8,7,14,25)(2,10,11,17,13,12,3,16)(4,19,20,21,23,26,22,27)(9,28,29,18,30,31,15,32)Polycyclic presentation:
2^9|a^2=eh,b^2=e,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214424, #72214675, #72214756 |
| maximal normal subgroup of | #72214424, #72214675, #72214756 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408090);