Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
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 | |
16873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(14,16);(1,2)(3,4)(5,6)(7,8)(10,12)(14,16);(9,10)(11,12)(13,14,15,16);(5,6)(7,8);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^9|c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #1202, #72216629, #72216649, … |
| maximal normal subgroup of | #1202, #72216629, #72219257, … |
| maximal quotient of | #72241162, #72241305, #72250275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519655);