Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,12)(13,15)(14,16)(18,22)(19,23)(25,26)(27,31)(28,32)(29,30);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,22)(23,24)(25,27,29,32)(26,28,30,31);(2,7)(4,6)(10,15)(11,16)(18,22)(20,24)(26,30)(27,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fh,b^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232588, #72232591, #72232595, … |
| maximal normal subgroup of | #72232588, #72232591, #72232595, … |
| maximal quotient of | #72328544 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4523613);