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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1974 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)(5,6,11,16)(7,19,21,28)(8,18,12,14)(9,24,26,35)(10,27,29,33)(13,32,22,15)(17,23,34,30)(20,36,31,25);(5,7)(6,13)(8,20)(9,14)(10,24)(11,21)(12,31)(15,30)(16,22)(17,25)(18,26)(19,27)(23,32)(28,33)(29,35)(34,36);(2,4)(5,8,11,12)(6,14,16,18)(7,20,21,31)(9,22,26,13)(10,23,29,30)(15,24,32,35)(17,33,34,27)(19,25,28,36);(5,9)(6,15)(7,17)(8,22)(10,28)(11,26)(12,13)(14,24)(16,32)(18,35)(19,29)(20,33)(21,34)(23,36)(25,30)(27,31);(6,16)(7,21)(14,18)(15,32)(17,34)(20,31)(24,35)(27,33)Polycyclic presentation:
2^9|a^2=i,c^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,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320537, #72322566, #72343374, … |
| maximal normal subgroup of | #72320537, #72322566, #72343374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530825);