Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#120 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1975 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,12,21)(13,23,24,17)(15,22,16,20);(1,3)(2,5)(4,7)(6,8)(9,11,14,12)(10,15,18,16)(13,22,24,20)(17,21,23,19);(5,8);(9,12,14,11)(10,16,18,15)(13,20,24,22)(17,19,23,21)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,e^d=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72303518, #72303526, #72303531, … |
| maximal normal subgroup of | #72303518, #72303526, #72303531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155659);