Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4066 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)(3,13)(4,18)(5,12)(6,10)(7,11)(8,15)(9,19)(14,23)(16,25)(17,24)(20,27)(21,26)(22,31)(28,30)(29,32);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,23)(10,24)(11,25)(12,13)(18,32)(19,30)(20,31)(21,29)(22,26)(27,28);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,27)(11,26)(12,18)(13,28)(15,30)(16,29)(17,31)(23,25)(24,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,23)(10,18)(11,26)(12,27)(13,28)(15,29)(16,30)(17,31)(22,25)(24,32)Polycyclic presentation:
2^9|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,h^a=hi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72215968, #72215974, … |
| maximal normal subgroup of | #72215953, #72215968, #72215974, … |
| maximal quotient of | #72231835, #72234089, #72234095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301302);