Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1695 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)(7,14)(8,10)(9,18)(11,13)(12,16)(15,19)(17,20);(5,6,11,18,10,8,9,13)(7,15,19,14,17,12,16,20);(1,2,3,4)(5,7,10,17)(6,12,8,15)(9,19,11,16)(13,20,18,14);(5,8,10,6)(7,15,17,12)(9,13,11,18)(14,19,20,16)Polycyclic presentation:
2^9|b^2=ei,c^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338468, #72338487, #72338490, … |
| maximal normal subgroup of | #72338468, #72338487, #72338490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330418);