Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1972 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)(7,15)(8,19)(9,24)(10,13)(11,18)(12,22)(14,21)(16,34)(17,33)(20,31)(23,32)(25,29)(26,30)(27,36)(28,35);(5,7)(6,11)(8,20)(9,25)(10,17)(12,29)(13,28)(14,31)(15,19)(16,21)(18,22)(23,36)(24,27)(26,35)(30,34)(32,33);(2,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,18)(16,27)(17,28)(19,22)(20,29)(21,24)(23,30)(25,31)(26,32)(33,35)(34,36);(5,8)(6,12)(7,16)(9,26)(10,23)(11,27)(13,30)(14,32)(15,33)(17,25)(18,35)(19,24)(21,22)(28,31);(6,13)(7,17)(12,30)(14,24)(16,25)(18,35)(19,32)(20,36)Polycyclic presentation:
2^9|f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343457, #72343825, #72343827, … |
| maximal normal subgroup of | #72343457, #72343825, #72343827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530843);