Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188814 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,20,24,21,23,29,31)(19,25,26,32,27,22,28,30);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16);(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30);(3,11)(7,15)(10,16)(12,13)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72305449, #72305453, #72305485, … |
| maximal normal subgroup of | #72305449, #72305453, #72305485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96114);