Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16508 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,16,23,17,24,15,8)(4,18,19,26,21,25,20,10)(14,22,28,27,29,32,30,31);(1,3)(2,8,12,23)(4,14)(5,15)(6,16)(7,17)(9,24,11,13)(10,22,26,32)(18,27,25,31)(19,28)(20,30)(21,29);(2,9)(4,19)(5,6)(8,13)(10,26)(11,12)(14,28)(15,16)(20,21)(23,24)(27,31)(29,30);(1,4)(2,10)(3,14)(5,20)(6,19)(7,21)(8,22)(9,25)(11,18)(12,26)(13,27)(15,30)(16,28)(17,29)(23,32)(24,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72215972, #72216013, … |
| maximal normal subgroup of | #72213514, #72215972, #72216013, … |
| maximal quotient of | #72250593, C2×#464656, #72252125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371852);