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 | C2×D42 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2094 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,29)(23,31,33,27)(24,28,26,30)(32,35,34,36);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,33)(22,27,29,31)(24,32,26,34)(28,36,30,35);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(25,26)(29,30)(32,34)(35,36);(6,12)(10,13)(11,18)(17,19)(22,29)(27,31)(28,30)(35,36);(7,15)(10,13)(14,20)(17,19)(22,29)(23,32)(24,26)(27,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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 | #72311027, #72311030, #72311050, … |
| maximal normal subgroup of | #72311027, #72311030, #72311050, … |
| maximal quotient of | #72311428, #72311719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576686);