Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16)(8,12)(9,14,24,13)(10,15)(11,18,26,22)(17,31)(19,30)(20,28)(21,27,23,29)(25,33,36,34)(32,35);(1,2)(3,4)(5,7,10,19)(6,11,15,26)(8,17,20,32)(9,22,24,18)(12,25,28,36)(13,16,14,30)(21,34,23,33)(27,35,29,31);(2,4)(5,8)(6,12)(7,17)(9,23)(10,20)(11,25)(13,29)(14,27)(15,28)(16,31)(18,34)(19,32)(21,24)(22,33)(26,36)(30,35);(6,13)(8,20)(9,24)(11,16)(12,27)(14,15)(17,32)(18,22)(25,35)(26,30)(28,29)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72286038, #72286067, #72286156, … |
| maximal normal subgroup of | #72286038, #72286067, #72286156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408287);