Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110532 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15,19,23)(12,14)(13,17)(16,20)(18,21,24,22);(1,3)(2,5,6,7)(4,8)(10,14,17,16)(15,21,23,22);(9,11)(10,15,17,23)(12,18)(13,19)(14,21,16,22)(20,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^c=eh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234285, #72258988, #72259001, … |
| maximal normal subgroup of | #72234285, #72258988, #72259001, … |
| maximal quotient of | #72234300, C2×#110532 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17728);