Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
5298 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)(5,6,7,8)(13,14)(15,19)(16,20)(17,18);(10,12)(14,18)(16,19);(1,2,4,3)(9,10)(11,12)(14,18)(15,20);(5,6)(7,8)(13,15)(14,16)(17,20)(18,19);(1,3,4,2)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72277955, C2×#1099602, #72280907, … |
| maximal normal subgroup of | #72277955, C2×#1099602, #72280907, … |
| maximal quotient of | #72278390, #72278768, #72279031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1006798);