Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1986 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)(22,28)(23,32)(25,26)(29,33)(30,31)(35,36);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,22,25,28,27,31,26,30)(23,24,29,35,33,34,32,36);(7,13)(10,17)(14,19)(18,20)(21,23,26,32,27,33,25,29)(22,24,30,36,31,34,28,35);(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,30)(25,35)(26,36)(27,34)(28,33)(31,32);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,33)(24,34)(29,32)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308331 |
| maximal normal subgroup of | #72308331 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404929);