Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,19,20,17)(21,22,25,28)(23,24,27,26);(8,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,23)(22,26)(24,28)(25,27);(6,7)(10,11)(15,20)(17,19)(23,27)(24,26);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,15)(14,17)(16,20)(18,19)(21,24,25,26)(22,27,28,23)Polycyclic presentation:
2^9|a^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300470, #72300513, #72300543 |
| maximal normal subgroup of | #72300470, #72300513, #72300543 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5385948);