Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,20,24,22)(25,26)(27,29)(28,30)(31,32);(2,6)(4,7)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,23)(21,24)(25,26,28,30)(27,29,31,32);(2,6)(3,8)(10,14)(13,15)(17,19,21,24)(18,22,23,20)(25,27,28,31)(26,29,30,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,31)(30,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326557, #72326579, #72326613, … |
| maximal normal subgroup of | #72326557, #72326579, #72326613, … |
| maximal quotient of | #72328480 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1024723);