Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7832 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)(15,16)(17,19)(18,20);(2,4)(6,9)(10,12);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20);(2,4)(14,19)(15,20);(13,16,17,18)(14,15,19,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72234935, #72237366, #72238183, … |
| maximal normal subgroup of | #72234935, #72237366, #72238183, … |
| maximal quotient of | #72239397, #72262085, #72304056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540696);