Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1614 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)(5,6,9,8)(7,10,11,12)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,28)(30,32)(31,37)(33,41)(43,44);(7,11)(10,12)(14,19)(15,20)(21,28)(22,25)(29,30,34,32)(31,35,37,39)(33,40,41,36)(38,44,42,43);(5,7,9,11)(6,10,8,12)(13,15,18,20)(14,16,19,24)(17,25,27,22)(21,26,28,23)(29,31)(30,35)(32,39)(33,42)(34,37)(36,44)(38,41)(40,43);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,16)(14,20)(15,19)(17,26)(18,24)(21,22)(23,27)(25,28)(29,32,34,30)(31,35,37,39)(33,36,41,40)(38,44,42,43);(5,8,9,6)(7,12,11,10)(13,17,18,27)(14,21,19,28)(15,22,20,25)(16,23,24,26)(29,33,34,41)(30,36,32,40)(31,38,37,42)(35,43,39,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4067234);