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 | |
7623 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)(3,5)(4,6)(7,8)(17,18)(19,23)(20,22)(21,24);(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,19,21,20)(18,22,24,23);(2,3)(6,7)(10,13)(14,15)(19,20)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,24,22);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,22,24,23);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^9|b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232276, #72232278, #72232577, … |
| maximal normal subgroup of | #72232276, #72232278, #72232577, … |
| maximal quotient of | #72232300, #72232331, #72232338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7640487);