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 | |
6407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,12)(11,16,13,15)(17,18)(19,20)(21,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,12,16);(1,3)(2,5)(4,8)(6,7)(9,10,14,12)(11,16,13,15)(17,19,22,21)(18,23,24,20);(17,20,22,23)(18,19,24,21);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,16,13,15)(17,19,22,21)(18,20,24,23);(1,2)(3,5)(4,7)(6,8)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,23,24,20)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233523, #72233539, #72233585, … |
| maximal normal subgroup of | #72233523, #72233539, #72233585, … |
| maximal quotient of | #72297740, #72297774, #72297930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592882);