Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394275 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2135 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)(5,6)(7,13,15,19)(8,11)(9,12)(10,16,14,17)(18,20);(5,6)(7,14,15,10)(8,11)(9,12)(13,16,19,17)(18,20);(5,7)(6,10)(8,17)(9,15)(11,19)(12,14)(13,20)(16,18);(2,4)(8,18)(11,20)(13,19)(16,17)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232947, #72233091, #72233100, … |
| maximal normal subgroup of | #72232947, #72233091, #72233100, … |
| maximal quotient of | #72232921, #72233090, #72236944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301668);