Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,5,8,14,16)(3,10,12,13,11,7,15,4)(18,22,25,30)(19,27)(20,28)(23,24,31,26)(29,32);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,18,21,25)(19,23,29,31)(20,24,28,26)(22,27,30,32);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,19,28,32,21,29,20,27)(18,23,26,22,25,31,24,30);(3,11)(4,13)(7,10)(12,15)(17,20,21,28)(18,24,25,26)(19,27,29,32)(22,23,30,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285730, #72285760, #72287271, … |
| maximal normal subgroup of | #72285730, #72285760, #72287271, … |
| maximal quotient of | #72285843, #72285913, #72286937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391344);