Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187419 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62841 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(2,7,9,12)(8,15,16,14)(17,18)(19,27,30,23)(20,31,32,40)(21,26)(22,25)(24,33,38,34)(28,43)(29,42,47,37)(35,39)(36,45)(41,46)(44,48);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,48)(32,44)(33,45)(34,46)(35,47)(38,41)(39,42)(40,43);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,20)(18,24)(19,28)(21,34)(22,32)(23,36)(25,38)(26,40)(27,41)(29,46)(30,44)(31,39)(33,35)(37,43)(42,48)(45,47);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(18,25)(20,32)(23,27)(26,39)(28,44)(33,34)(37,42)(45,46)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235011, #72235012, C2×#187423, … |
| maximal normal subgroup of | #72235011, #72235012, C2×#187423, … |
| maximal quotient of | C2×#187423, #72239243, #72239246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94619);