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 | |
6358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(9,10,13,14)(11,15,16,12)(17,18,22,23)(19,27,28,24)(20,29,30,25)(21,31,32,26);(1,2,4,6)(3,5,7,8)(10,14)(12,15)(18,23)(21,32)(24,27)(25,29);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(18,23)(19,28)(25,29)(26,31);(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,25)(19,26)(21,24)(22,30)(23,29)(27,32)(28,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,26)(19,25)(20,24)(22,32)(23,31)(27,30)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261925, #72262081, #72326449, … |
| maximal normal subgroup of | #72261925, #72262081, #72326449, … |
| maximal quotient of | #72326539, #72326540, #72326769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658969);