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 | |
6513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(4,8)(6,7)(9,10,14,12)(11,15,13,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(18,21)(22,24);(17,18)(19,22)(20,21)(23,24);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,15,12,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72233523, #72233536, #72233544, … |
| maximal normal subgroup of | #72233523, #72233536, #72233544, … |
| maximal quotient of | #72261952, #72297839, #72297860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7584498);