Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,32)(28,40)(29,34)(30,43)(31,35)(33,41)(36,51)(37,46)(38,47)(39,48)(42,49)(44,50)(45,53)(52,56)(54,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,36)(29,38)(30,37)(31,39)(33,45)(34,47)(35,48)(40,51)(41,53)(42,54)(43,46)(44,52)(49,55)(50,56);(1,4,5,3)(2,6,7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,36)(29,41)(30,44)(31,42)(32,45)(34,49)(35,50)(37,52)(38,53)(39,54)(40,43)(46,51)(47,55)(48,56);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,29,31,30)(26,34,35,43)(27,37,39,38)(28,41,42,44)(32,46,48,47)(33,49,50,40)(36,52,54,53)(45,51,56,55);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,43)(27,38,39,37)(28,41,42,44)(32,47,48,46)(33,49,50,40)(36,53,54,52)(45,55,56,51)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6475088);