Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38916 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,26,30)(23,27,33,36)(24,28,25,29)(31,34,32,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,23)(22,27)(24,32)(25,31)(26,33)(28,35)(29,34)(30,36);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24)(22,28)(25,26)(29,30)(31,32)(34,35);(6,13)(11,15)(12,14)(19,20);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,31)(24,25)(27,34)(28,29)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284360, #72284796, #72285334, … |
| maximal normal subgroup of | #72284360, #72284796, #72285334, … |
| maximal quotient of | #72311783, #72311930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581172);