Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#465 |
| 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 |
| prime radical | D4×#121 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2804 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)(14,17)(15,19)(18,20);(5,7)(6,9)(8,12)(10,11)(13,14,15,17,16,18,19,20);(2,4)(7,12)(9,10);(5,7)(6,10)(8,12)(9,11);(7,12)(9,10)Polycyclic presentation:
2^9|b^2=fh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=eh110 conjugacy classes — view the full character table.
| maximal subgroup of | #72282779, #72283069, #72283082, … |
| maximal normal subgroup of | #72282779, #72283069, #72283082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503912);