Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418512 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1686 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)(4,16)(5,21)(6,23)(7,11)(8,14)(9,19)(10,22)(13,24)(15,28)(17,29)(18,27)(20,26)(25,31)(30,32);(1,3)(2,8)(4,13)(5,6)(7,15)(9,25)(10,28)(11,21)(12,23)(14,22)(16,32)(17,18)(19,30)(20,31)(24,27)(26,29);(1,4)(2,9)(3,13)(5,17)(6,18)(7,20)(8,24)(10,26)(11,27)(12,29)(14,30)(15,31)(16,23)(19,22)(21,25)(28,32);(1,5,22,15,7,14,6,3)(2,8,23,28,11,21,10,12)(4,17,19,31,20,30,18,13)(9,24,16,32,27,25,26,29)Polycyclic presentation:
2^9|d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72215936, #72216059, … |
| maximal normal subgroup of | #72213603, #72215936, #72216059, … |
| maximal quotient of | C2×#418513, #72252186, #72252201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325709);