Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13995 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,11,7,3,6,14)(4,15,18,9,19,25,17,12)(5,20,21,10,23,26,22,13)(16,27,29,28,30,24,31,32);(1,3,7,2)(4,12)(5,13,23,10)(6,11,8,14)(9,19)(15,18)(16,27)(17,25)(20,22,26,21)(24,30)(28,29)(31,32);(1,4)(2,9)(3,12)(5,16)(6,17)(7,19)(8,18)(10,24)(11,25)(13,27)(14,15)(20,28)(21,29)(22,31)(23,30)(26,32);(1,5)(2,10)(3,13)(4,16)(6,22)(7,23)(8,21)(9,24)(11,26)(12,27)(14,20)(15,28)(17,31)(18,29)(19,30)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72216141, #72219685, … |
| maximal normal subgroup of | #72215953, #72216141, #72219685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273934);