Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| 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 | #33746 | C2 |
| 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(18,22,24,31)(19,26)(20,27)(23,29,32,25)(28,30);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(17,18)(19,23)(20,29)(21,24)(22,30)(25,27)(26,31)(28,32);(1,4,10,14,6,13,5,12)(2,7,8,15,3,11,9,16)(17,19,27,30,21,28,20,26)(18,23,25,22,24,32,29,31);(2,3)(7,11)(8,9)(15,16)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216707, #72216730, #72216771, … |
| maximal normal subgroup of | #72216707, #72216730, #72216771, … |
| maximal quotient of | #72225053, #72225095, #72227152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391153);