Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,14,17,19)(8,12)(9,18)(10,13)(11,15,20,16);(5,7,8,16,10,17,9,15)(6,11,12,19,13,20,18,14);(7,15,17,16)(11,19,20,14);(2,4)(7,16,17,15)(11,19,20,14)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219666, #72219673, … |
| maximal normal subgroup of | #72216067, #72219666, #72219673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339371);