Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16508 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,14,22,17,24,16,8)(4,18,19,26,21,25,20,10)(15,23,28,27,29,31,30,32);(1,3,5,16,7,17,6,14)(2,8,11,13,12,22,9,24)(4,15,20,30,21,29,19,28)(10,23,18,27,26,31,25,32);(2,9)(3,14)(4,19)(5,6)(8,22)(10,26)(11,12)(15,29)(16,17)(20,21)(23,27)(31,32);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,23)(9,25)(11,18)(12,26)(13,27)(14,28)(16,30)(17,29)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^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,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338622, #72338624, #72338629 |
| maximal normal subgroup of | #72338622, #72338624, #72338629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371860);