Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196134 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2930 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)(5,6,9,13,10,12,8,17)(7,11,14,18,15,19,16,20);(2,4)(5,7,9,14,10,15,8,16)(6,11,13,18,12,19,17,20);(6,12)(7,14)(8,9)(11,18)(15,16)(19,20);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72215972, #72216071, … |
| maximal normal subgroup of | #72215520, #72215972, #72216071, … |
| maximal quotient of | #72238750, C2×#364468, #72248313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271664);