Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2662 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)(7,14,16,17)(11,15,19,20);(2,4)(5,6)(7,15,16,20)(8,18)(9,12)(10,13)(11,17,19,14);(5,7)(6,11)(8,17)(9,14)(10,16)(12,20)(13,19)(15,18);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,20)(15,17)(16,19)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219653, #72219666, … |
| maximal normal subgroup of | #72216067, #72219653, #72219666, … |
| maximal quotient of | #72233799, #72233810, #72233845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271718);