Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3416 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,22)(10,12)(14,19)(15,20)(16,26)(17,24)(18,32)(21,23)(25,33)(27,31)(28,30)(29,34)(35,36);(6,11)(8,19)(9,20)(12,23)(13,24)(14,15)(17,31)(18,30)(21,33)(22,25)(26,34)(27,28)(29,36)(32,35);(1,3,4,2)(6,12)(11,23)(13,26)(14,25)(15,22)(24,34)(27,35)(28,32);(5,7)(6,13)(8,17)(9,18)(10,16)(11,24)(12,26)(14,27)(15,28)(19,31)(20,30)(21,29)(22,32)(23,34)(25,35)(33,36);(7,16)(13,26)(17,29)(18,30)(24,34)(27,35)(28,32)(31,36)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=hi,h^2=i,b^a=bf,c^a=ci,f^a=fh,h^a=hi,f^b=fh,h^b=hi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72322712, #72322724, #72322732, … |
| maximal normal subgroup of | #72322712, #72322724, #72322732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529814);