Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1251 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)(7,11)(8,18)(9,14)(10,13)(12,20)(15,28)(16,25)(17,24)(19,31)(21,32)(22,27)(23,29)(26,34)(30,36)(33,35);(1,2,4,3)(5,7,10,17)(6,11,13,24)(8,19,21,15)(9,16,22,30)(12,23,26,35)(14,25,27,36)(18,31,32,28)(20,29,34,33);(5,8,9,20)(6,12,14,18)(7,15,16,29)(10,21,22,34)(11,23,25,28)(13,26,27,32)(17,19,30,33)(24,35,36,31);(1,3,4,2)(6,13)(11,24)(12,26)(14,27)(18,32)(23,35)(25,36)(28,31)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312509, #72312510, #72312517, … |
| maximal normal subgroup of | #72312509, #72312510, #72312517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410385);