Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7513948 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2760 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1759 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)(3,11,14,7)(4,15,18,8)(5,9)(6,10)(12,25)(13,21,27,23)(16,20)(17,28,30,22)(19,24)(26,31)(29,32);(1,3)(2,7)(4,16)(5,13)(6,14)(8,20)(9,23)(10,11)(12,18)(15,25)(17,29)(19,27)(21,24)(22,32)(26,30)(28,31);(1,4)(2,8)(3,12,14,16)(5,17)(6,18)(7,20,11,25)(9,22)(10,15)(13,26,27,29)(19,30)(21,31,23,32)(24,28);(1,5)(2,9)(3,13)(4,17)(6,19)(7,21)(8,22)(10,24)(11,23)(12,26)(14,27)(15,28)(16,29)(18,30)(20,31)(25,32);(5,19)(9,24)(13,27)(17,30)(21,23)(22,28)(26,29)(31,32)Polycyclic presentation:
2^9|a^2=gh,c^2=fg,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,g^b=gi,h^c=hi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237728, #72237732, #72237743, … |
| maximal normal subgroup of | #72237728, #72237732, #72237743, … |
| maximal quotient of | #72284397, #72284424, #72284460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421144);