Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464193 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2657 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16502 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1718 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)(3,9)(4,16)(5,8)(6,11)(7,12)(10,20)(13,26)(14,23)(15,25)(17,24)(18,28)(19,27)(21,22)(29,32)(30,31);(2,8,12,11)(9,23,25,22)(10,16,28,27)(24,32,30,26);(1,3)(2,9)(4,17)(5,21)(6,14)(7,15)(8,22)(10,26)(11,23)(12,25)(13,18)(16,30)(19,31)(20,29)(24,27)(28,32);(1,4,5,18,7,19,6,20)(2,10,11,27,12,28,8,16)(3,13,14,17,15,29,21,31)(9,24,22,26,25,30,23,32)Polycyclic presentation:
2^9|b^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279455, #72284943, #72287274, … |
| maximal normal subgroup of | #72279455, #72284943, #72287274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371389);