Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,17,18)(15,21,22,19)(16,20,25,24)(23,27,28,26)(29,30)(31,32)(33,37)(34,35)(36,40)(38,39)(41,44)(42,43);(6,9)(10,12)(14,18)(15,22)(20,24)(23,28)(30,35)(31,36)(37,43)(38,41);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,27)(26,28)(29,31,34,36)(30,32,35,40)(33,41,42,38)(37,44,43,39);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,32)(30,36)(31,35)(33,39)(34,40)(37,41)(38,43)(42,44);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28)(29,33,34,42)(30,37,35,43)(31,38,36,41)(32,39,40,44)Polycyclic presentation:
2^9|a^2=g,c^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4574224);