Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(10,12)(14,15)(17,18,21,20)(19,22,24,23)(25,26)(27,30)(28,29)(31,32);(2,4)(3,7)(9,10,13,12)(11,15,16,14)(17,19,21,24)(18,22,20,23)(25,27)(26,30)(28,31)(29,32);(2,4)(6,8)(10,12)(11,16)(18,20)(22,23)(25,27,29,32)(26,31,28,30);(1,3,5,7)(2,6,4,8)(9,11,13,16)(10,14,12,15)(17,20,21,18)(19,23,24,22)(25,27)(26,31)(28,30)(29,32);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=gi,b^2=fgi,c^2=i,d^2=gh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300946, #72326559, #72326588, … |
| maximal normal subgroup of | #72300946, #72326559, #72326588, … |
| maximal quotient of | #72328396, #72328909, #72329118 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5110701);