Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #62824 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3486 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,17,25)(3,8,14,23,13,21,28,31)(4,7,16,20,15,9,18,24)(12,19,27,30,26,22,29,32);(2,7)(4,15)(5,17)(8,19)(9,11)(10,20)(12,26)(14,28)(21,22)(23,30)(24,25)(31,32);(8,21)(19,22)(23,31)(30,32);(1,3)(2,8)(4,12)(5,14)(6,13)(7,19)(9,22)(10,23)(11,21)(15,26)(16,27)(17,28)(18,29)(20,30)(24,32)(25,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72260900, #72261107, #72261176, … |
| maximal normal subgroup of | #72260900, #72261107, #72261176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400682);