Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504667 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1957 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)(3,9)(4,18)(5,21)(6,11)(7,12)(8,24)(10,23)(13,14)(15,27)(16,28)(17,29)(19,25)(20,22)(26,31)(30,32);(1,3,6,15,7,17,8,16)(2,9,24,28,12,29,11,27)(4,19,20,30,10,26,21,13)(5,14,23,31,22,32,18,25);(1,4,7,10)(2,5,12,22)(3,13)(6,21,8,20)(9,25)(11,23,24,18)(14,27)(15,26)(16,19)(17,30)(28,32)(29,31);(1,5)(2,10)(3,14)(4,12)(6,18)(7,22)(8,23)(9,26)(11,21)(13,27)(15,25)(16,31)(17,32)(19,29)(20,24)(28,30);(1,6,7,8)(2,11,12,24)(3,15,17,16)(4,20,10,21)(5,18,22,23)(9,27,29,28)(13,19,30,26)(14,25,32,31)Polycyclic presentation:
2^9|b^2=fg,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216029, #72216047, … |
| maximal normal subgroup of | #72215721, #72216029, #72216047, … |
| maximal quotient of | #72250774, C2×#7504669, #72255596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411865);