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 | |
9318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20);(2,4)(6,9)(10,12);(1,2)(3,4)(13,15,16,20)(14,17,18,19);(14,18)(17,19);(2,4)(15,20)(17,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|c^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 | #72267392, #72267396, #72279258, … |
| maximal normal subgroup of | #72267392, #72267396, #72279258, … |
| maximal quotient of | #72268518, C2×#7634984, #72280075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542180);