Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1702 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)(5,6,10,8)(7,11,9,12)(13,14,17,18)(15,21,22,19)(16,20,24,26)(23,28,25,27)(29,30)(31,36)(32,38)(33,41)(34,35)(37,42)(39,40)(43,44);(5,7,10,9)(6,11,8,12)(14,18)(15,22)(16,24)(27,28)(30,35)(31,38)(33,42)(40,43);(5,8,10,6)(7,11,9,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,31,34,38)(30,36,35,32)(33,39,42,44)(37,43,41,40);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,15)(14,19)(16,25)(17,22)(18,21)(20,28)(23,24)(26,27)(29,32,34,36)(30,31,35,38)(33,43,42,40)(37,44,41,39);(13,16)(14,20)(15,23)(17,24)(18,26)(19,27)(21,28)(22,25)(29,33,34,42)(30,37,35,41)(31,39,38,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=gh,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2984632);