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 | |
6485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,5)(3,6,8,7)(9,10)(11,16)(12,15)(13,14);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,21)(22,23);(2,5)(6,7)(10,14)(11,15)(18,23)(19,20);(17,19,22,20)(18,24,23,21);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,20,22,19)(18,21,23,24);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,14)(13,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=gi,f^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 | #72233544, #72233573, #72233593, … |
| maximal normal subgroup of | #72233544, #72233573, #72233593, … |
| maximal quotient of | #72297813, #72297829, #72297912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644622);