Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62983 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(9,10,14,18)(11,19,22,23)(12,17,13,16)(15,21,24,20);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,23,17,19);(1,3)(2,5)(4,7)(6,8);(10,16)(12,13)(15,23)(17,18)(19,24)(20,21)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72238184, #72238188, #72238193, … |
| maximal normal subgroup of | #72238184, #72238188, #72238193, … |
| maximal quotient of | #72240201, #72240223, #72240291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253723);