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 | |
3598 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)(7,11)(8,9)(10,12)(14,18)(15,20)(22,24)(23,27);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,18)(19,20)(23,27)(26,28);(6,8)(10,11)(14,18)(16,19)(21,22,25,24)(23,26,27,28);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,15,17,20)(14,19,18,16)(21,23)(22,26)(24,28)(25,27);(5,8,9,6)(7,11,12,10)(13,16,17,19)(14,15,18,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|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=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72327241, #72327249, #72327298, … |
| maximal normal subgroup of | #72327241, #72327249, #72327298, … |
| maximal quotient of | #72333073 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3386357);