Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418818 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15266 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,14,7,3,8,11)(4,12,16,25,17,9,23,26)(5,18,20,10,22,19,21,13)(15,28,29,24,30,32,31,27);(1,3)(2,7)(4,9)(5,19)(6,11)(8,14)(10,21)(12,17)(13,20)(15,28)(16,26)(18,22)(23,25)(24,29)(27,31)(30,32);(1,4)(2,9)(3,12)(5,15)(6,23)(7,17)(8,16)(10,24)(11,26)(13,27)(14,25)(18,32)(19,28)(20,31)(21,29)(22,30);(1,5)(2,10)(3,13)(4,15)(6,21)(7,22)(8,20)(9,24)(11,19)(12,27)(14,18)(16,31)(17,30)(23,29)(25,32)(26,28)Polycyclic presentation:
2^9|a^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72263012, #72267112, #72338514, … |
| maximal normal subgroup of | #72263012, #72267112, #72338514, … |
| maximal quotient of | #72262979 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326046);