Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| 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 | #42883 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,32)(28,38)(29,34)(30,43)(31,35)(33,36)(37,46)(39,51)(40,45)(41,49)(42,53)(44,48)(47,50)(52,55)(54,56);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,30)(26,32,35,43)(28,36,40,44)(29,37,41,42)(33,45,48,38)(34,46,49,53)(39,50,52,54)(47,55,56,51);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,40)(26,33,35,48)(27,36,30,44)(29,39,41,52)(32,45,43,38)(34,47,49,56)(37,50,42,54)(46,55,53,51);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,37)(28,39)(30,42)(31,41)(32,46)(33,47)(35,49)(36,50)(38,51)(40,52)(43,53)(44,54)(45,55)(48,56);(2,5)(6,8)(11,18)(14,22)(17,19)(21,23)(29,41)(34,49)(37,42)(39,52)(46,53)(47,56)(50,54)(51,55)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7164710);