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 | |
5970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,32)(28,31);(9,11)(10,14)(12,15)(13,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32);(2,5)(6,7)(10,15)(12,14)(18,24)(19,20)(26,29)(27,28);(1,2)(3,6)(4,5)(7,8)(9,12,13,14)(10,11,15,16)(17,20,22,19)(18,25,24,23)(21,27,30,28)(26,31,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,21,22,30)(18,26,24,29)(19,27,20,28)(23,31,25,32);(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^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 | #72312021, #72318304, #72318421, … |
| maximal normal subgroup of | #72312021, #72318304, #72318421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7684075);