Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15267 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2254 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)(3,7)(4,14,15,8)(5,17)(6,10)(9,20)(11,25)(12,22,29,30)(13,23)(16,24,19,26)(18,21)(27,32,28,31);(3,11,13,18)(7,21,23,25)(12,28,29,27)(22,32,30,31);(1,3)(2,7)(4,12)(5,18)(6,13)(8,22)(9,25)(10,23)(11,20)(14,30)(15,29)(16,27)(17,21)(19,28)(24,31)(26,32);(1,4)(2,8)(3,12)(5,19)(6,15)(7,22)(9,26)(10,14)(11,27)(13,29)(16,20)(17,24)(18,28)(21,31)(23,30)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72216141, #72219482, … |
| maximal normal subgroup of | #72215953, #72216141, #72219482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326261);