Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
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 | |
4030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,18,21,22)(19,24,23,20)(26,27)(30,31);(2,5)(6,8)(10,13)(11,16)(25,26)(27,29)(28,31)(30,32);(11,16)(14,15)(18,22)(19,23)(26,27)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,14,13,15)(17,19,21,23)(18,20,22,24)(25,27,29,26)(28,31,32,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,22)(21,24)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=fgh,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232588, #72232615, #72232658, … |
| maximal normal subgroup of | #72232588, #72232615, #72232658, … |
| maximal quotient of | #72328453, #72330153 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5864972);