Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
3407 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)(9,10,12,13)(11,15,16,14)(17,18,21,22)(19,23,24,20)(26,27)(28,32);(2,3)(4,8)(10,13)(14,15)(17,18)(19,23)(20,24)(21,22)(26,27)(30,31);(2,3)(6,7)(11,16)(14,15)(18,22)(19,24)(25,26,29,27)(28,30,32,31);(1,3,5,2)(4,6,8,7)(9,11,12,16)(10,14,13,15)(17,19,21,24)(18,20,22,23)(25,27,29,26)(28,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=f,c^2=fghi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72327607, #72327637, #72327749, … |
| maximal normal subgroup of | #72327607, #72327637, #72327749, … |
| maximal quotient of | #72330713 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3236734);