Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3836 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)(3,4)(5,7)(6,8)(10,14)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(18,22)(19,24)(26,29)(30,31);(2,7)(3,8)(10,14)(11,15)(17,19,21,24)(18,23,22,20)(25,27)(26,30)(28,32)(29,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,29)(30,32);(1,4)(2,8)(3,7)(5,6)(9,12)(10,15)(11,14)(13,16)(17,20,21,23)(18,24,22,19)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|c^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72328573, #72328647, #72328664 |
| maximal normal subgroup of | #72328573, #72328647, #72328664 |
| maximal quotient of | #72330255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2666401);