Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6428 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)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,20)(17,19);(2,4)(6,10)(9,12);(2,4);(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|b^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237567, #72238000, #72239649, … |
| maximal normal subgroup of | #72237567, #72238000, #72239649, … |
| maximal quotient of | #72239783, #72315906, #72315959 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347512);