Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(7,13,15,17)(10,19,14,20);(5,6)(7,14,15,10)(8,16)(9,12)(11,18)(13,19,17,20);(5,7)(6,10)(8,17)(9,15)(11,20)(12,14)(13,18)(16,19);(2,4)(5,6)(7,10)(8,11)(9,12)(13,19)(14,15)(16,18)(17,20)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233091, #72237986, #72261123, … |
| maximal normal subgroup of | #72233091, #72237986, #72261123, … |
| maximal quotient of | #72233076, #72237980 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325167);