Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2670 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,7,8)(13,14,17,16)(15,18,19,20);(10,12)(14,16)(18,20);(1,2,4,3)(9,10)(11,12);(5,6)(7,8)(13,15)(14,18)(16,20)(17,19);(1,3,4,2)(13,16,17,14)(15,18,19,20)Polycyclic presentation:
2^9|a^2=fi,c^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72327795, #72327805, #72327817, … |
| maximal normal subgroup of | #72327795, #72327805, #72327817, … |
| maximal quotient of | #72331386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1334402);