Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18)(19,25)(20,28,29,23)(21,24)(22,27)(26,30,32,31);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,22,24,25)(20,26)(21,27)(23,30,28,31)(29,32);(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,f^b=fh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312160, #72312175, #72312251, … |
| maximal normal subgroup of | #72312160, #72312175, #72312251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13011);