Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42440 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,29)(24,27)(25,30)(26,28)(31,34)(32,35)(33,36);(1,2)(3,4)(7,13)(8,16)(10,17)(11,15)(21,23)(22,27)(24,25)(26,32)(28,36)(29,30)(31,35)(33,34);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(22,28)(23,24)(26,34)(27,35)(29,36)(30,31);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24,25,23)(22,29,30,27)(26,33,34,32)(28,35,31,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,30)(23,24)(28,31)(32,33)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7073166);