Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14019 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1338 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)(3,8,14,27)(4,16,19,21)(5,9,20,18)(6,11)(7,10)(12,30)(13,25)(15,26,23,28)(17,24)(22,29)(31,32);(1,3,7,23,6,14,22,15)(2,8,10,28,11,27,29,26)(4,17,18,12,19,31,9,25)(5,13,21,24,20,30,16,32);(1,4,7,18,6,19,22,9)(2,5,10,21,11,20,29,16)(3,12,15,17,14,25,23,31)(8,24,26,13,27,32,28,30);(1,5)(2,9)(3,13)(4,10)(6,20)(7,21)(8,25)(11,18)(12,27)(14,30)(15,32)(16,22)(17,28)(19,29)(23,24)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215938, #72219684, #72219691 |
| maximal normal subgroup of | #72215938, #72219684, #72219691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275669);