Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1785 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)(5,6,10,8)(7,11,9,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30)(31,37)(32,41)(33,42)(34,35)(36,40)(38,39)(43,44);(5,7,10,9)(6,11,8,12)(16,24)(19,26)(22,28)(25,27)(30,35)(33,41)(36,37)(39,43);(5,8,10,6)(7,11,9,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,31,34,40)(30,36,35,37)(32,38,42,44)(33,39,41,43);(1,2)(3,4)(5,7,10,9)(6,12,8,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32,34,42)(30,33,35,41)(31,38,40,44)(36,39,37,43);(5,9,10,7)(6,11,8,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,33)(30,32)(31,39)(34,41)(35,42)(36,38)(37,44)(40,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5545372);