Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7266801 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #46996 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1691 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,21)(13,20)(15,19)(16,23)(17,22)(18,25)(24,27)(26,30)(28,31)(29,32);(1,3)(2,7,10,11)(4,15,17,12)(5,13)(6,14)(8,21)(9,24,25,20)(16,28,29,26)(18,27)(19,22)(23,30)(31,32);(5,18)(7,11)(8,22)(9,25)(13,27)(16,29)(26,28)(30,31);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,23)(10,22)(11,21)(13,26)(14,15)(18,29)(20,30)(24,31)(25,32)(27,28);(1,5,6,18)(2,9,10,25)(3,13,14,27)(4,16,17,29)(7,20,11,24)(8,23,22,32)(12,26,15,28)(19,30,21,31)Polycyclic presentation:
2^9|b^2=fgh,e^2=i,b^a=bf,c^a=cg,e^a=ef,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237743, #72237815, #72237867, … |
| maximal normal subgroup of | #72237743, #72237815, #72237867, … |
| maximal quotient of | C2×#7266802, #72283474, #72283499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7173998);