Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6379185 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,16,18,8)(5,19)(6,10)(7,15)(9,20)(12,26,28,25)(13,23)(14,22)(17,24,32,31)(21,29,30,27);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,22)(10,23)(11,19)(15,20)(16,30)(17,27)(18,28)(24,26)(25,31)(29,32);(1,4)(2,8)(3,12)(5,17)(6,18)(7,21)(9,24)(10,16)(11,25)(13,27)(14,28)(15,29)(19,31)(20,32)(22,26)(23,30);(3,13)(5,20)(7,22)(9,19)(11,23)(12,27)(14,15)(17,32)(21,26)(24,31)(25,30)(28,29);(3,14)(7,23)(11,22)(12,28)(13,15)(21,30)(25,26)(27,29)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,e^b=ei,f^b=fi,h^c=hi,f^d=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72236873, #72237542, #72237550, … |
| maximal normal subgroup of | #72236873, #72237542, #72237550, … |
| maximal quotient of | #72237649, C2×#6379185, #72251086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286381);