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 | |
6082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,32)(30,31);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,29)(21,31)(22,28)(24,27)(25,32)(26,30);(2,6)(4,8)(9,10,12,14)(11,13,15,16)(18,24)(19,28)(25,31)(26,29);(2,6)(3,7)(11,15)(13,16)(17,20)(18,25)(19,29)(21,27)(22,30)(23,32)(24,31)(26,28);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,26)(19,25)(20,23)(22,32)(24,29)(27,30)(28,31);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,27)(20,29)(21,25)(22,28)(23,24)(26,30)(31,32)Polycyclic presentation:
2^9|c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249434, #72249476, #72249496, … |
| maximal normal subgroup of | #72249434, #72249476, #72249496, … |
| maximal quotient of | #72262630, #72262635, #72262805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805295);