Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498890 |
| 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 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14)(7,11,16,26)(8,18,20,30)(9,21,24,13)(12,23,28,19)(15,31,32,36)(17,29,22,27)(25,34,35,33);(1,2)(3,4)(5,7)(6,11)(8,15)(9,22)(10,16)(12,25)(13,29)(14,26)(17,24)(18,31)(19,33)(20,32)(21,27)(23,34)(28,35)(30,36);(1,3,4,2)(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,30)(14,28)(16,32)(17,34)(18,21)(19,24)(22,33)(26,35)(27,36)(29,31);(7,16)(11,26)(15,32)(17,22)(25,35)(27,29)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=i,c^2=e,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,f^c=fi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286038, #72286046, #72286056, … |
| maximal normal subgroup of | #72286038, #72286046, #72286056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406330);