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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,26)(25,27)(30,31)(32,39)(33,41)(43,44);(7,11)(9,12)(14,18)(19,21)(20,24)(27,28)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,44,42,43);(5,7)(6,9)(8,11)(10,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,23)(22,28)(25,27)(29,31,34,30)(32,35,39,37)(33,36,41,38)(40,44,42,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,32)(30,35)(31,37)(33,42)(34,39)(36,44)(38,43)(40,41);(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=g,d^2=fg,e^2=f,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, 4039474);