Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,8)(6,7)(10,11)(12,16)(20,27)(24,29)(25,31)(30,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(1,2,5,3)(4,6,8,7)(9,10,13,11)(12,15,16,14)(18,22)(23,28)(24,29)(30,32)(35,42)(38,41)(43,48)(45,47);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35)(34,38)(36,43)(37,42)(39,45)(40,41)(44,48)(46,47);(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,37,44)(34,39,40,46)(35,43,42,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=fgi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363879);