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 | |
5625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,8)(9,10)(11,15)(12,13)(14,16);(2,6)(3,7)(9,10,13,12)(11,14,16,15)(17,18)(19,20)(21,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24);(2,6)(3,7)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,22)(18,19,23,24);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256306, #72263008, #72263018 |
| maximal normal subgroup of | #72256306, #72263008, #72263018 |
| maximal quotient of | #72300867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10051008);