Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682564 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,9)(7,11)(13,14,17,19)(15,16,20,18);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,20);(2,4)(13,15,18,14,17,20,16,19);(5,7,8,11)(6,10,9,12)(14,19)(15,20);(2,4)(6,9)(10,12)(13,16,17,18)(14,20,19,15)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282672, #72282693, #72282952, … |
| maximal normal subgroup of | #72282672, #72282693, #72282952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590835);