Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
12076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2)(3,4)(6,8)(9,10,14,12)(11,16,13,15);(5,6,7,8)(9,11,14,13)(10,15,12,16);(2,4)(9,10,14,12)(11,16,13,15);(1,2)(3,4);(9,12,14,10)(11,16,13,15);(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,e^2=i,f^2=i,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 | #72262454, #72262464, #72262494, … |
| maximal normal subgroup of | #72262454, #72262464, #72262494, … |
| maximal quotient of | C2×#7689777, #72281131, #72281364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7596973);