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 | |
5786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,16)(12,14)(13,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(1,2)(3,6)(4,7)(5,8)(9,11,13,14)(10,12,15,16)(20,27)(23,25)(28,32)(30,31);(1,3)(2,5)(4,8)(6,7)(18,19)(23,25)(24,26)(30,31);(9,11,13,14)(10,12,15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,2)(3,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72327258, #72327643 |
| maximal normal subgroup of | #72327258, #72327643 |
| maximal quotient of | #72327366, #72327774, #72327900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9576131);