Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483919 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #2184 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,19,25,21,23,27,26)(20,29,28,24,31,22,30,32);(3,9)(6,12)(10,15)(13,16)(18,22)(19,27)(20,30)(23,29)(24,25)(26,32)(28,31);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,19,21,27)(18,23)(20,30,31,28)(25,26);(2,7)(6,12)(8,14)(13,16)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284874, #72285695, #72286391, … |
| maximal normal subgroup of | #72284874, #72285695, #72286391, … |
| maximal quotient of | #72343328, #72343329, #72343337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391120);