Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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:
(3,7)(6,8)(10,13)(11,15)(17,18)(19,24)(20,22)(21,23)(25,26)(27,28)(29,30)(31,32);(1,2,5,4)(3,6,7,8)(9,10)(11,16)(12,13)(14,15)(17,19,22,21)(18,23,20,24)(26,30)(27,31);(11,15)(14,16)(18,20)(23,24)(25,27,29,31)(26,28,30,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,24,21,23)(25,27)(26,28)(29,31)(30,32);(1,4,5,2)(3,8,7,6)(9,10)(11,16)(12,13)(14,15)(17,21,22,19)(18,23,20,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|b^2=fghi,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72258884, #72258888, #72258889, … |
| maximal normal subgroup of | #72258884, #72258888, #72258889, … |
| maximal quotient of | #72331344, #72331383, #72334904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712065);