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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3760 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,20,16,18)(21,22,25,27)(23,24,28,26);(6,7)(10,11)(13,15,17,16)(14,18,19,20)(21,23,25,28)(22,26,27,24);(8,12)(10,11)(14,19)(18,20)(22,27)(23,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,17,15)(14,20,19,18)(21,24,25,26)(22,28,27,23);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,d^2=f,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 | #72281123, #72281351, #72282095, … |
| maximal normal subgroup of | #72281123, #72281351, #72282095, … |
| maximal quotient of | #72332778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2016594);