Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3407 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)(3,14,17,30)(4,10)(5,20,23,11)(6,12)(7,13)(8,16)(9,15,27,18)(19,24,22,29)(21,26)(25,31)(28,32);(1,3)(2,8)(4,18)(5,21)(6,15)(7,17)(9,24)(10,28)(11,30)(12,25)(13,26)(14,20)(16,23)(19,31)(22,32)(27,29);(1,3)(2,9)(4,15)(5,16)(6,18)(7,17)(8,24)(10,14)(11,25)(12,30)(13,27)(19,31)(20,28)(21,23)(22,32)(26,29);(1,4,7,6)(2,10,13,12)(3,15,17,18)(5,19,23,22)(8,25,26,28)(9,14,27,30)(11,29,20,24)(16,31,21,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72287178, C2×#493422, #72287234, … |
| maximal normal subgroup of | #72287178, C2×#493422, #72287234, … |
| maximal quotient of | C2×#493422, #72318066, #72318246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400618);