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 | |
7146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,15)(11,12,16,14)(17,18)(19,20)(21,24)(22,23);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,22)(23,24);(2,3)(6,7)(10,15)(12,14)(17,20)(18,21)(19,23)(22,24);(4,8)(6,7)(10,15)(11,16)(17,19,22,21)(18,20,23,24);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,14)(15,16)(17,21,22,19)(18,20,23,24);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232519, #72241094, #72241131, … |
| maximal normal subgroup of | #72232519, #72241094, #72241131, … |
| maximal quotient of | #72232628, #72304043, #72304130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8519481);