Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201664 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,15,20,23)(12,16)(13,17)(14,18)(19,22,21,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15,18,23)(12,19)(13,21)(14,20)(16,22,17,24);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24);(9,13,14,12)(10,17,18,16)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72236820, #72236823, #72251232, … |
| maximal normal subgroup of | #72236820, #72236823, #72251232, … |
| maximal quotient of | #72263703, #72263719, #72263728 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108863);