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 | |
1646 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)(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,18)(17,24)(20,25)(21,27)(22,23)(26,28)(29,30,34,35)(31,32,38,36)(33,37,42,43)(39,40,44,41);(5,7,9,6)(8,10,12,11)(14,16)(19,23)(20,24)(26,27)(30,35)(31,38)(37,43)(39,44);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,31,34,38)(30,36,35,32)(33,39,42,44)(37,41,43,40);(1,2)(3,4)(13,16,18,14)(15,19,22,23)(17,20,25,24)(21,27,28,26)(29,32,34,36)(30,31,35,38)(33,41,42,40)(37,44,43,39);(13,17,18,25)(14,20,16,24)(15,21,22,28)(19,26,23,27)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,41)(38,44)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,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, 2669753);