Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,19)(15,20,16,18);(2,4)(6,9)(10,12);(1,2)(3,4)(13,14)(15,18)(16,20)(17,19);(13,15,17,16)(14,18,19,20);(2,4)(13,16,17,15)(14,18,19,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72234903, #72234904, #72237366, … |
| maximal normal subgroup of | #72234903, #72234904, #72237366, … |
| maximal quotient of | #72303648, #72303656, #72304054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542318);