Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421159 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,13,8,12,9,17)(7,14,18,20,16,11,15,19);(2,4)(7,15,16,18)(11,19,14,20);(1,2,3,4)(5,7,8,16)(6,11)(9,18,10,15)(12,14)(13,20)(17,19);(6,12)(9,10)(11,14)(15,18)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338559, #72338561, #72338563, … |
| maximal normal subgroup of | #72338559, #72338561, #72338563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342032);