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×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,15)(8,11)(9,14)(10,13)(12,18)(16,17)(19,20);(1,2)(3,4)(5,6,10,14,11,8,9,13)(7,16,18,20,19,12,17,15);(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,15)(14,20);(2,4)(5,8,11,6)(7,16,19,12)(9,13,10,14)(15,17,20,18)Polycyclic presentation:
2^9|b^2=f,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 | #72227999, #72228012, #72228014 |
| maximal normal subgroup of | #72227999, #72228012, #72228014 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327973);