Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(6,8)(10,12)(13,14,16,18)(15,17,19,20);(1,2,4,3)(15,19)(17,20);(9,10)(11,12)(13,15)(14,17)(16,19)(18,20);(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=f,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 | #72264082, #72264110, D4×#2405, … |
| maximal normal subgroup of | #72264082, #72264110, D4×#2405, … |
| maximal quotient of | #72307672, #72334306, C2×#612624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519820);