Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1774 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)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,18)(17,24)(20,25)(21,23)(22,28)(26,27)(29,30,34,36)(31,37,38,32)(33,41,42,35)(39,40,44,43);(5,7,8,11)(6,9,10,12)(14,16)(15,21)(20,24)(22,27)(31,38)(32,37)(39,44)(40,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,21)(14,19,16,23)(17,22,25,27)(20,26,24,28)(29,31)(30,32)(33,39)(34,38)(35,40)(36,37)(41,43)(42,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,21,23)(17,20,25,24)(22,28,27,26)(29,32)(30,31)(33,40)(34,37)(35,39)(36,38)(41,44)(42,43);(13,17,18,25)(14,20,16,24)(15,22,21,27)(19,26,23,28)(29,33,34,42)(30,35,36,41)(31,39,38,44)(32,40,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668792);