Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454037 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16249 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2193 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)(3,12)(4,17)(5,10)(6,11)(7,15)(8,19)(9,20)(13,23)(14,27)(16,24)(18,25)(21,26)(22,30)(28,31)(29,32);(1,3)(2,7)(4,14)(5,13)(6,16)(8,22)(9,24)(10,12)(11,23)(15,20)(17,31)(18,30)(19,29)(21,28)(25,27)(26,32);(3,13)(4,18)(5,20)(7,12)(8,25)(9,10)(14,29)(15,16)(17,26)(19,21)(23,24)(27,32);(1,4)(2,8)(3,14)(5,21)(6,19)(7,22)(9,26)(10,25)(11,17)(12,27)(13,28)(15,30)(16,29)(18,20)(23,31)(24,32)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216013, #72216047, … |
| maximal normal subgroup of | #72215958, #72216013, #72216047, … |
| maximal quotient of | C2×#454037, #72255766, #72265741 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361233);