Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6,8,7)(13,14,17,18)(15,19,20,16);(1,2)(3,4)(10,12)(14,18)(15,20);(5,6,8,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19);(1,2,3,4)(5,7,8,6)(13,15)(14,19)(16,18)(17,20);(1,2)(3,4)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=gi,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
| maximal subgroup of | #72261002, #72261566, #72261780, … |
| maximal normal subgroup of | #72261002, #72261566, #72261780, … |
| maximal quotient of | #72298488, #72335950, #72336005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 427518);