Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 | |
2048 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)(6,13)(7,17)(8,21)(9,12)(10,25)(14,32)(15,35)(16,31)(18,22)(19,27)(23,29)(26,28)(30,34)(33,36);(5,6,11,16)(7,18,17,14)(8,22,23,32)(9,15,12,28)(10,26,27,35)(13,19,31,25)(20,34,24,33)(21,36,29,30);(2,4)(5,7)(6,14)(8,19)(9,24)(10,21)(11,17)(12,20)(13,22)(15,34)(16,18)(23,25)(26,30)(27,29)(28,33)(31,32)(35,36);(5,8)(6,14)(7,19)(9,21)(10,24)(11,23)(12,29)(13,30)(15,33)(16,18)(17,25)(20,27)(22,26)(28,34)(31,36)(32,35);(5,9,11,12)(6,15,16,28)(7,20,17,24)(8,21,23,29)(10,19,27,25)(13,26,31,35)(14,33,18,34)(22,36,32,30)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216688, #72224996, #72225012, … |
| maximal normal subgroup of | #72216688, #72224996, #72225012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530840);