Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1787 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)(24,34)(25,27)(26,33)(28,31)(30,35)(32,36);(1,2)(3,4)(7,13)(8,16)(10,17)(11,15)(21,22)(23,30)(24,28)(25,27)(26,32)(29,35)(31,34)(33,36);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,31,35,25,32,24,33)(22,26,34,36,27,30,28,29);(2,4)(7,13)(10,17)(14,19)(18,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,32)(31,34)(35,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,27)(23,32)(28,34)(33,35)Polycyclic presentation:
2^9|a^2=hi,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7080828);