Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14064 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,5,12,6,11)(3,14,16,28,17,8,22,25)(4,18,20,9,21,10,19,27)(15,29,31,23,30,24,32,26);(1,3,6,16,5,17,7,22)(2,8,13,28,12,14,11,25)(4,15,20,32,21,30,19,31)(9,26,18,29,27,23,10,24);(1,4)(2,9)(3,15)(5,21)(6,20)(7,19)(8,23)(10,11)(12,27)(13,18)(14,26)(16,32)(17,30)(22,31)(24,28)(25,29);(1,4)(2,10,12,18)(3,15)(5,21)(6,20)(7,19)(8,24,14,29)(9,11,27,13)(16,32)(17,30)(22,31)(23,28,26,25)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915, #72339024, #72339028, … |
| maximal normal subgroup of | #72279915, #72339024, #72339028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278224);