Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(18,22)(19,25)(20,27)(30,32)(33,34,37,41)(35,39,43,38)(36,40,46,45)(42,48,44,47);(10,14)(15,16)(18,22)(19,25)(24,29)(26,31)(33,35,37,43)(34,38,41,39)(36,42,46,44)(40,47,45,48);(1,2,5,4)(3,6,7,8)(9,10,13,14)(11,15,12,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,39)(36,44)(37,43)(38,41)(40,47)(42,46)(45,48);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,39)(36,45)(37,41)(38,43)(40,46)(42,47)(44,48);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,45)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3597684);