Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3084 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)(5,6)(7,12)(8,10)(9,11)(14,15)(16,20)(22,25)(23,27);(1,2,4,3)(5,7)(6,8)(9,10)(11,12)(13,14,17,15)(16,18,20,19)(23,27)(26,28);(2,3)(7,10)(8,12)(13,14,17,15)(16,19,20,18)(21,22)(23,26)(24,25)(27,28);(5,6)(7,12)(8,10)(9,11)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(1,3,4,2)(5,8,9,12)(6,10,11,7)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|b^2=fghi,c^2=fh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72331973, #72331979, #72332058, … |
| maximal normal subgroup of | #72331973, #72331979, #72332058, … |
| maximal quotient of | #72333165 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2075463);