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 | |
7148 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)(17,18,21,23)(19,20,24,22);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19)(18,22)(20,23)(21,24);(2,4)(3,7)(9,10,12,13)(11,14,15,16)(18,23)(20,22);(3,7)(5,8)(11,15)(14,16)(17,19,21,24)(18,22,23,20);(1,4,6,2)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24);(1,5)(2,3)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=g,e^2=gh,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 | #72241094, #72241131, #72241138, … |
| maximal normal subgroup of | #72241094, #72241131, #72241138, … |
| maximal quotient of | #72304139, #72336595, #72336720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506865);