Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1558 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)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,25)(16,31,33,36)(18,29)(20,28,24,30)(23,27)(26,32,35,34);(1,2)(3,4)(5,7)(6,11)(8,16)(9,23)(10,18)(12,26)(13,29)(14,27)(15,22)(17,25)(19,35)(20,34)(21,33)(24,32)(28,31)(30,36);(2,4)(8,20,21,24)(12,28,19,30)(16,32,33,34)(26,36,35,31);(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,30)(14,19)(15,31)(17,34)(18,33)(20,25)(22,28)(23,32)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72262990 |
| maximal normal subgroup of | #72262990 |
| maximal quotient of | #72266586, #72266588, #72266644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364784);