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 | |
7137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,16)(11,14,12,15)(17,18,21,22)(19,23,24,20);(1,2,5,4)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,24)(22,23);(2,4)(6,8)(9,10)(11,15)(12,14)(13,16)(18,22)(20,23);(3,7)(6,8)(9,10,13,16)(11,14,12,15)(17,20)(18,19)(21,23)(22,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72232576, #72233990, #72238559, … |
| maximal normal subgroup of | #72232576, #72233990, #72238559, … |
| maximal quotient of | #72232596, #72233998, #72234005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7686861);