Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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×#3220 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,35)(32,36)(33,34);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,27)(24,32)(25,33)(26,31)(28,34)(29,36)(30,35);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(22,28)(24,25)(27,34)(29,30)(32,33)(35,36);(6,13)(11,15)(12,14)(19,20)(23,31)(27,35)(32,33)(34,36);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,24,26,25)(22,29,30,28)(23,32,31,33)(27,36,35,34)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72307362, #72307611, #72307864, … |
| maximal normal subgroup of | #72307362, #72307611, #72307864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582369);