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 | |
12180 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)(6,8)(12,16)(14,15);(5,6,7,8)(9,10)(11,13)(12,14)(15,16);(2,3)(10,11)(14,15);(1,3,4,2);(9,11,13,10)(12,15,16,14);(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234475, #72234484, #72262329, … |
| maximal normal subgroup of | #72234475, #72234484, #72262329, … |
| maximal quotient of | #72262140, #72262149, #72297618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7584315);