Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| 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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1962 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)(6,12)(7,16)(8,10)(9,21)(13,27)(14,30)(15,33)(17,29)(18,35)(19,26)(20,24)(22,25)(28,32)(31,34);(5,6)(7,17)(8,15)(9,22)(10,25)(11,13)(12,21)(14,18)(16,34)(19,30)(20,27)(23,31)(24,33)(26,28)(29,35)(32,36);(2,4)(6,13)(12,27)(14,29)(15,25)(17,30)(22,33)(28,34)(31,32);(5,7)(6,14)(8,16)(9,23)(10,26)(11,19)(12,28)(13,29)(15,31)(17,22)(18,21)(20,35)(24,36)(25,32)(27,34)(30,33);(5,8,11,10)(6,15,13,25)(7,16,19,26)(9,20,24,21)(12,22,27,33)(14,31,29,32)(17,34,30,28)(18,23,35,36)Polycyclic presentation:
2^9|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,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320537 |
| maximal normal subgroup of | #72320537 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530813);