Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3283 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,27)(17,20)(18,23)(21,28)(22,30)(24,37)(25,31)(26,34)(29,36)(32,38)(33,40)(35,41)(39,43)(42,44);(6,9)(7,11)(13,15,18,26)(14,19,23,34)(16,24,28,38)(17,25,30,40)(20,31,22,33)(21,32,27,37)(29,39,41,44)(35,42,36,43);(14,20)(17,30)(19,31)(21,35)(22,23)(25,40)(27,36)(29,41)(32,42)(33,34)(37,43)(39,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,29)(18,28)(19,32)(20,35)(22,36)(23,27)(25,39)(26,38)(30,41)(31,42)(33,43)(34,37)(40,44);(1,3,4,2)(6,9)(10,12)(16,28)(21,27)(24,38)(29,41)(32,37)(35,36)(39,44)(42,43)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306934);