Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461124 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16380 |
| 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 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,12)(3,13,21,25,15,8,16,24)(4,17,19,27,20,9,22,28)(14,29,31,32,30,23,18,26);(1,3)(2,8,11,13)(4,18)(5,21)(6,16)(7,15)(9,26,17,32)(10,24,12,25)(14,19)(20,31)(22,30)(23,28,29,27);(1,4)(2,9)(3,14)(5,22)(6,19)(7,20)(8,23)(10,27)(11,17)(12,28)(13,29)(15,30)(16,31)(18,21)(24,32)(25,26);(2,10)(5,6)(8,24)(9,27)(11,12)(13,25)(14,30)(16,21)(17,28)(19,22)(23,26)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251788, #72251791, #72251794, … |
| maximal normal subgroup of | #72251788, #72251791, #72251794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368320);