Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,4,7,6)(11,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(34,38)(35,41)(36,44)(46,48);(3,7)(4,6)(18,22)(20,29)(23,25)(26,32)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,46,47,48);(1,3,5,7)(2,6,8,4)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,5,6)(2,7,8,3)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,2)(3,6)(4,7)(5,8)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=gi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4401967);