Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1652 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)(13,14)(15,22)(16,20)(17,25)(18,19)(21,27)(23,24)(26,28)(29,30,34,38)(31,35,36,32)(33,37,42,44)(39,41,43,40);(6,9)(10,12)(14,19)(15,20)(21,25)(23,28)(29,31,34,36)(30,32,38,35)(33,39,42,43)(37,40,44,41);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,26)(27,28)(29,31)(30,35)(32,38)(33,39)(34,36)(37,41)(40,44)(42,43);(1,2)(3,4)(13,16)(14,20)(15,19)(17,26)(18,22)(21,23)(24,27)(25,28)(29,32,34,35)(30,36,38,31)(33,41,42,40)(37,39,44,43);(5,7,8,11)(6,10,9,12)(13,17,18,27)(14,21,19,25)(15,23,20,28)(16,24,22,26)(29,33)(30,37)(31,39)(32,40)(34,42)(35,41)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,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, 2646078);