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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1251 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)(25,26,30,35)(27,31,38,34)(28,39,41,43)(29,33,42,48)(32,45,47,36)(37,44,51,49)(40,52,53,54)(46,55,56,50);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,35,34)(28,36,41,45)(29,37,42,51)(32,43,47,39)(33,44,48,49)(40,50,53,55)(46,54,56,52);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,27,36,30,41,38,45)(26,32,34,39,35,47,31,43)(29,40,37,50,42,53,51,55)(33,46,49,52,48,56,44,54);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,29)(26,33)(27,37)(28,40)(30,42)(31,44)(32,46)(34,49)(35,48)(36,50)(38,51)(39,52)(41,53)(43,54)(45,55)(47,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(29,42)(33,48)(37,51)(40,53)(44,49)(46,56)(50,55)(52,54)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fg,d^2=f,e^2=f,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, 6959599);