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 | |
1670 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)(15,22,20,16)(17,21,25,26)(23,28,27,24)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(14,19)(15,20)(17,25)(24,28)(29,31,34,32)(30,35,37,38)(33,40,42,39)(36,44,41,43);(5,7,10,9)(6,11,8,12)(13,15,18,20)(14,16,19,22)(17,23,25,27)(21,24,26,28)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,26)(25,28)(29,30)(31,35)(32,38)(33,36)(34,37)(39,43)(40,44)(41,42);(5,9,10,7)(6,11,8,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)(29,33,34,42)(30,36,37,41)(31,39,32,40)(35,43,38,44)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=gh,e^2=fg,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, 4843848);