Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
24673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 16 points:
(2,4)(6,8);(5,6)(7,8)(10,12)(14,16);(1,2)(3,4)(5,6)(7,8)(9,10,13,12)(11,14,15,16);(11,15)(14,16);(9,11)(10,14)(12,16)(13,15);(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72232070, #120×H(5,𝔽2), #72233748, … |
| maximal normal subgroup of | #72232070, #120×H(5,𝔽2), #72233748, … |
| maximal quotient of | #72232130, #72232532, #72233551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540234);