Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(22,24);(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,22,23,24);(2,5)(6,8)(10,11)(12,16);(3,7)(6,8)(12,16)(14,15);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,19,20,23)(18,22,21,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|b^2=g,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72300691, #72342294, #72342328, … |
| maximal normal subgroup of | #72300691, #72342294, #72342328, … |
| maximal quotient of | #72300745 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976742);