Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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)(9,10,12,13)(11,15,16,14)(17,18,21,22)(19,20,23,24)(25,26)(27,31)(28,32)(29,30);(2,3)(6,7)(9,10)(11,14)(12,13)(15,16)(18,22)(19,23)(26,30)(28,31);(4,8)(6,7)(10,13)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27,29,32)(26,31,30,28);(1,3,5,2)(4,6,8,7)(9,11,12,16)(10,14,13,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,22)(21,24)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fgi,c^2=gi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72278309, #72300745, #72300755, … |
| maximal normal subgroup of | #72278309, #72300745, #72300755, … |
| maximal quotient of | #72278856, #72278926, #72299565 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3457938);