Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,11)(14,16)(15,20)(21,22)(23,28)(24,27)(25,26);(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,19,20,18)(21,23)(22,24)(25,28)(26,27);(13,15)(14,18)(16,19)(17,20)(21,24)(22,27)(23,25)(26,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,27)(26,28);(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,19,20,18)(21,25,26,22)(23,28,27,24)Polycyclic presentation:
2^9|a^2=fi,b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300838, #72300855, #72300900, … |
| maximal normal subgroup of | #72300838, #72300855, #72300900, … |
| maximal quotient of | #72328754 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327809);