Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16632 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,21,9)(3,12,14,7,16,27,15,23)(4,17,18,8,20,26,19,25)(13,24,29,28,31,22,30,32);(1,3)(2,7,10,23)(4,13)(5,14)(6,16)(8,24,25,22)(9,12,11,27)(15,21)(17,32,26,28)(18,29)(19,30)(20,31);(1,4)(2,8,10,25)(3,13)(5,18)(6,20)(7,22,23,24)(9,17,11,26)(12,28,27,32)(14,29)(15,30)(16,31)(19,21);(2,9)(4,18)(5,21)(7,12)(8,25)(10,11)(13,29)(14,15)(19,20)(22,24)(23,27)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216130, #72216142, … |
| maximal normal subgroup of | #72216037, #72216130, #72216142, … |
| maximal quotient of | C2×#468843, #72252107, #72265531 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376039);