Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
3729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,27)(26,28);(1,2,3,4)(6,7)(10,11)(13,14,16,18)(15,17,19,20)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(8,12)(10,11)(15,19)(17,20)(22,27)(24,26);(5,7,9,6)(8,10,12,11)(13,14)(15,20)(16,18)(17,19)(21,23)(22,24)(25,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,24)(22,28)(23,27)(25,26)Polycyclic presentation:
2^9|b^2=fghi,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72297732, #72297734, #72297772, … |
| maximal normal subgroup of | #72297732, #72297734, #72297772, … |
| maximal quotient of | #72298194, #72298195, #72330829 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3405099);