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 | |
6567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,7)(9,10,13,16)(11,12,14,15)(17,18)(19,23)(20,21)(22,24);(1,2)(3,7)(4,5)(6,8)(9,11)(10,12)(13,14)(15,16);(1,2,4,5)(3,6,8,7)(9,12)(10,14)(11,16)(13,15)(18,21)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,2)(3,7)(4,5)(6,8)(9,11)(10,12)(13,14)(15,16)(17,19,20,24)(18,22,21,23);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,16)(13,15)Polycyclic presentation:
2^9|a^2=gi,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72233544, #72233582, #72233593, … |
| maximal normal subgroup of | #72233544, #72233582, #72233593, … |
| maximal quotient of | #72257911, #72297849, #72297859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875377);