Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,16,18)(15,19,20,17);(5,7,9,10)(6,8,11,12)(13,15,16,20)(14,17,18,19);(2,4)(5,6)(7,12)(8,10)(9,11);(1,2)(3,4);(5,6)(7,8)(9,11)(10,12);(5,8)(6,10)(7,11)(9,12)Polycyclic presentation:
2^9|a^2=gi,b^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72257495, #72257697, #72262496, … |
| maximal normal subgroup of | #72257495, #72257697, #72262496, … |
| maximal quotient of | C2×#7690367, #72280809, #72281130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7597563);