Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,8)(10,12)(13,14,17,20)(15,16,19,18);(1,2,3,4)(10,12);(9,10,11,12)(13,15,17,19)(14,18,20,16);(1,2,3,4)(5,6)(7,8)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(13,16,17,18)(14,19,20,15)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72264569, #72279836, #72279903, … |
| maximal normal subgroup of | #72264569, #72279836, #72279903, … |
| maximal quotient of | #72298436, #72298522, C2×#783592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 690788);