Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,40)(36,47)(37,45)(39,51)(41,52)(42,48)(43,49)(46,53)(50,56)(54,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,43,42)(33,38,47,52)(34,45,49,48)(39,50,54,53)(46,51,56,55);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,47,44,52)(29,39,42,53,43,54,37,50)(34,46,45,51,49,56,48,55);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,37)(28,39)(30,42)(31,43)(32,45)(33,46)(35,49)(36,50)(38,51)(40,53)(41,54)(44,48)(47,56)(52,55);(10,16)(11,18)(15,24)(17,19)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,42,43,37)(33,38,47,52)(34,48,49,45)(39,53,54,50)(46,55,56,51)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6948280);