Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| 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 |
| prime radical | #25773 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1234 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)(7,15)(8,19)(9,13)(10,14)(11,17)(12,22)(16,30)(18,29)(20,34)(21,33)(23,27)(24,35)(25,32)(26,36)(28,31);(1,2,4,3)(5,7)(6,11)(8,20)(9,17)(10,18)(12,24)(13,15)(14,25)(16,21)(19,30)(22,31)(23,32)(26,28)(27,29)(33,34)(35,36);(5,8,10,22)(6,12,14,19)(7,16,18,31)(9,21,23,36)(11,24,25,34)(13,26,27,33)(15,28,29,30)(17,20,32,35);(1,3,4,2)(6,13)(7,17)(12,26)(14,27)(16,20)(18,32)(19,33)(31,35)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=ei,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312332, #72312351, #72312355, … |
| maximal normal subgroup of | #72312332, #72312351, #72312355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410435);