Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,15)(8,11)(9,13)(10,14)(12,16)(17,19)(18,20);(1,3,4,2)(6,11)(8,9)(12,15)(13,14)(16,18)(19,20);(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,15)(14,20);(5,7)(6,12)(8,18)(9,16)(10,17)(11,15)(13,19)(14,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72221412, C4×#23872, #72312221, … |
| maximal normal subgroup of | C4×#23872, #72312221, #72312238 |
| maximal quotient of | #72221412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271831);