Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13,15,8)(9,17,18,11)(10,12)(14,16)(19,20);(1,2,3,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13)(17,19)(18,20);(5,8)(6,7,12,15)(9,16)(10,13)(11,14,17,20)(18,19);(5,9)(6,11)(7,14)(8,16)(10,18)(12,17)(13,19)(15,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=ce,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72245877, #72245896, C4×#21776, … |
| maximal normal subgroup of | #72245877, #72245896, C4×#21776, … |
| maximal quotient of | #72251574, #72269859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262687);