Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,7)(8,12)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,34,35)(31,32,39,36)(33,37,41,42)(38,40,44,43);(5,6,9,7)(8,11,12,10)(15,22)(18,21)(23,28)(25,27)(30,35)(32,36)(33,41)(38,44);(6,7)(10,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31,34,39)(30,36,35,32)(33,38,41,44)(37,43,42,40);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,32,34,36)(30,31,35,39)(33,40,41,43)(37,38,42,44);(5,8,9,12)(6,10,7,11)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33)(30,37)(31,38)(32,40)(34,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4407984);