Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,17,20)(15,19,16,18)(22,25)(26,27);(5,6)(7,10)(8,9)(11,12)(21,22)(23,26)(24,25)(27,28);(5,6,8,9)(7,10,11,12)(13,15,17,16)(14,18,20,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=gh,c^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232306, #72232328, #72232419, … |
| maximal normal subgroup of | #72232306, #72232328, #72232419, … |
| maximal quotient of | #72323483 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1301045);