Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106380 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,20)(3,11)(4,16)(8,12,19,26)(10,24,23,21)(14,29)(15,25)(17,28)(18,30)(22,32,31,27);(1,2,4,10,13,19,28,31)(3,12,14,27,25,20,29,24)(5,8,17,22,6,9,16,23)(7,18,21,15,26,30,32,11);(3,11)(7,12)(14,18)(15,25)(20,26)(21,27)(24,32)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,e^b=eh,f^b=fi,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213205, #72213569, #72213628, … |
| maximal normal subgroup of | #72213205, #72213569, #72213628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13581);