Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470343 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,20)(15,16,19,17);(7,8)(10,11)(13,15,17,14,18,19,16,20);(2,4)(5,7,9,8)(6,10,12,11)(13,16,18,17)(14,19,20,15);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,19,20,15);(6,12)(10,11)(13,16,18,17)(14,15,20,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72305276, #72305277, #72305290, … |
| maximal normal subgroup of | #72305276, #72305277, #72305290, … |
| maximal quotient of | #72306139, #72306153, #72314436 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394952);