Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2684 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,18)(5,12)(6,11)(7,10)(8,16)(9,20)(14,26)(15,25)(17,24)(19,23)(21,28)(22,27)(29,31)(30,32);(1,3,6,15,7,17,5,16)(2,8,12,24,10,25,11,13)(4,19,20,30,21,14,22,31)(9,26,18,32,27,23,28,29);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,27)(11,18)(12,28)(13,29)(15,30)(16,31)(17,19)(24,32)(25,26);(3,15)(5,6)(8,24)(11,12)(13,25)(14,30)(16,17)(18,28)(19,31)(20,22)(23,32)(26,29);(2,10)(8,25)(9,27)(11,12)(13,24)(18,28)(23,26)(29,32)Polycyclic presentation:
2^9|b^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216142, #72216577, … |
| maximal normal subgroup of | #72215721, #72216142, #72216577, … |
| maximal quotient of | C2×#7504292, #72252096, #72265675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411488);