Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437473 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,15)(8,14)(9,13)(10,12)(11,16)(17,20)(18,19);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,19)(13,20)(14,15);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(2,4);(6,12)(11,19)(13,14)(15,20)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219483, #72233807, #72234089, … |
| maximal normal subgroup of | #72219483, #72233807, #72234089, … |
| maximal quotient of | #72213216, #72214707, #72219481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344669);