Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1758 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)(7,11)(10,12)(13,14,17,21)(15,19,23,18)(16,20,26,25)(22,28,24,27)(30,32)(31,37)(33,41)(43,44);(6,9)(10,12)(13,15,17,23)(14,18,21,19)(16,22,26,24)(20,27,25,28)(29,30,34,32)(31,35,37,38)(33,36,41,40)(39,43,42,44);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,27)(22,26)(25,28)(29,31)(30,35)(32,38)(33,42)(34,37)(36,44)(39,41)(40,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,25)(17,21)(18,23)(20,26)(22,27)(24,28)(29,32,34,30)(31,38,37,35)(33,36,41,40)(39,43,42,44);(13,16)(14,20)(15,22)(17,26)(18,27)(19,28)(21,25)(23,24)(29,33,34,41)(30,36,32,40)(31,39,37,42)(35,43,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3377936);