Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394481 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14725 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2129 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)(3,8)(4,16,17,25)(5,19)(6,10)(7,11)(9,18,24,20)(12,26)(13,28,29,31)(14,23)(15,21)(22,27,32,30);(3,12,14,15)(8,21,23,26)(13,27,29,30)(22,31,32,28);(1,3)(2,8)(4,13)(5,15)(6,12)(7,14)(9,22)(10,26)(11,23)(16,31)(17,29)(18,30)(19,21)(20,27)(24,32)(25,28);(1,4)(2,9)(3,13)(5,18)(6,20)(7,17)(8,22)(10,25)(11,24)(12,27)(14,29)(15,30)(16,19)(21,31)(23,32)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72216144, #72219688, … |
| maximal normal subgroup of | #72215974, #72216144, #72219688, … |
| maximal quotient of | C2×#395616 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302812);