Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10,13,15)(11,16,12,14)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,32)(30,31);(2,3)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,20)(19,22,24,23)(25,27)(26,29)(28,31)(30,32);(4,8)(6,7)(11,12)(14,16)(18,20)(22,23)(25,28,30,29)(26,27,31,32);(1,3,5,2)(4,7,8,6)(9,10)(11,16)(12,14)(13,15)(17,19,21,24)(18,22,20,23)(25,29,30,28)(26,27,31,32);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,14,15,16)(17,20,21,18)(19,23,24,22)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=hi,d^2=fgh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232606, #72232620, #72232644, … |
| maximal normal subgroup of | #72232606, #72232620, #72232644, … |
| maximal quotient of | #72328964, #72329163, #72329202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456874);