Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C42 |
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 | |
6368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,7)(13,14)(15,17)(16,19)(18,20);(10,12)(13,15,18,16)(14,19,20,17);(5,6,8,7)(9,10,11,12)(14,20)(15,16);(1,2,3,4)(5,7,8,6)(13,16,18,15)(14,19,20,17);(13,17,18,19)(14,15,20,16)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280835, #72280858, #72297607, … |
| maximal normal subgroup of | #72280835, #72280858, #72297607, … |
| maximal quotient of | #72298304 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 626897);