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 | |
13035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2,4,3)(5,6,7,8);(5,6)(7,8)(9,10,14,13)(11,16,12,15);(2,3)(9,11,14,12)(10,15,13,16);(1,3,4,2);(9,12,14,11)(10,15,13,16);(9,13,14,10)(11,16,12,15)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=h,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 | #72233538, #72233585, #72234479, … |
| maximal normal subgroup of | #72233538, #72233585, #72234479, … |
| maximal quotient of | #72261412, #72261932, #72297527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7595713);