Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1686 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)(5,6)(7,12)(8,10)(9,11)(13,14,18,19)(15,16,21,20)(17,22,25,26)(23,24,28,27)(29,30)(31,34)(32,40)(33,37)(35,39)(36,41)(38,43)(42,44);(14,19)(15,21)(17,25)(24,27)(29,31,33,39)(30,34,37,35)(32,38,41,44)(36,42,40,43);(5,7,10,9)(6,11,8,12)(13,15,18,21)(14,20,19,16)(17,23,25,28)(22,27,26,24)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,42)(41,44);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23)(29,30)(31,34)(32,36)(33,37)(35,39)(38,42)(40,41)(43,44);(5,9,10,7)(6,11,8,12)(13,17,18,25)(14,22,19,26)(15,23,21,28)(16,24,20,27)(29,32,33,41)(30,36,37,40)(31,38,39,44)(34,42,35,43)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843840);