Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 3145728 |
| 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:
(2,5)(6,8)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,30)(27,31)(29,32)(33,34,38,39)(35,41,42,36)(37,40,46,47)(43,45,48,44);(9,10,13,12)(11,15,16,14)(18,22)(19,26)(24,28)(27,29)(34,39)(35,42)(40,47)(43,48);(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,47,45);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,16,15)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,30)(28,31)(33,36)(34,35)(37,45)(38,41)(39,42)(40,48)(43,47)(44,46);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2696288);