Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
3442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,11)(9,10)(13,14,17,16)(15,19,20,18)(21,22)(23,25)(24,28)(26,27);(5,7,10,8)(6,11,9,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,27)(26,28);(6,9)(11,12)(13,14,17,16)(15,18,20,19)(21,24,26,25)(22,23,27,28);(1,2,3,4)(5,8,10,7)(6,12,9,11)(13,16,17,14)(15,18,20,19)(21,25,26,24)(22,23,27,28);(5,9,10,6)(7,12,8,11)(13,15)(14,19)(16,18)(17,20)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=fghi,e^2=gh,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 | #72327166, #72327249, #72327272, … |
| maximal normal subgroup of | #72327166, #72327249, #72327272, … |
| maximal quotient of | #72328811, #72332724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2236230);