Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6,9,8)(7,10,11,12)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(7,11)(10,12)(14,19)(15,22)(17,25)(24,27)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(1,2,4,3)(5,7)(6,10)(8,12)(9,11)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(1,3,4,2)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8,9,6)(7,12,11,10)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 477601);