Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| 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×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2561 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)(10,19)(13,20);(1,3,4,2)(5,6)(7,13)(8,16)(9,11)(10,15)(12,18)(14,20)(17,19);(5,7)(6,10)(8,17)(9,14)(11,19)(12,20)(13,16)(15,18);(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72221457, C4×#23882, #72252452, … |
| maximal normal subgroup of | C4×#23882, #72252452, #72252454, … |
| maximal quotient of | #72221457, #72257934, #72264027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271715);