Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #430064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15412 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,14,23)(4,16)(5,10)(6,9)(8,18)(11,20)(12,15,24,21)(13,29,30,28)(17,26)(19,27)(22,32,25,31);(1,3)(2,7,9,23)(4,13,17,30)(5,15)(6,14)(8,25)(10,24,11,12)(16,28)(18,31,19,32)(20,21)(22,27)(26,29);(1,2)(3,12)(4,8)(5,11)(6,9)(7,21)(10,20)(13,28)(14,24)(15,23)(16,18)(17,27)(19,26)(22,31)(25,32)(29,30);(1,4)(2,8)(3,13,14,30)(5,19)(6,17)(7,22,23,25)(9,27)(10,16)(11,26)(12,28,24,29)(15,32,21,31)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72215972, #72216071, … |
| maximal normal subgroup of | #72215958, #72215972, #72216071, … |
| maximal quotient of | C2×#430064, #72252130, #72314033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337260);