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 | |
6248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,22)(21,23);(1,2,6,5)(3,8,7,4)(17,19,21,20)(18,22,23,24);(1,3,6,7)(2,4,5,8)(10,13)(14,15)(18,23)(22,24);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23);(1,5,6,2)(3,4,7,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,19)(18,24,23,22);(1,4)(2,7)(3,5)(6,8)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72328779 |
| maximal normal subgroup of | #72328779 |
| maximal quotient of | #72328877, #72329122, #72329514 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7878899);