Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3514 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)(13,14,17,18)(15,19,20,16)(22,23)(24,28);(5,7,10,9)(6,11,8,12)(13,14)(15,19)(16,20)(17,18)(22,23)(26,27);(7,9)(11,12)(14,18)(16,19)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,15,17,20)(14,19,18,16)(21,23,25,22)(24,27,28,26);(5,9,10,7)(6,11,8,12)(13,16,17,19)(14,20,18,15)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fh,d^2=h,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 | #72332183, #72332227, #72332265 |
| maximal normal subgroup of | #72332183, #72332227, #72332265 |
| maximal quotient of | #72332764 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2046253);