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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,14,17,19)(15,16,20,18);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20);(2,4)(5,8,9,10)(6,11,12,7)(13,16,17,18)(14,15,19,20);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19);(5,7)(6,8)(9,11)(10,12);(5,6)(7,10)(8,11)(9,12)Polycyclic presentation:
2^9|a^2=gi,c^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72232032, #72232070, #72232285, … |
| maximal normal subgroup of | #72232032, #72232070, #72232285, … |
| maximal quotient of | #72232038, #72232109, #72232306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603240);