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 | |
1386 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)(27,35)(28,33)(29,31)(30,42)(32,38)(34,41)(36,48)(37,47)(39,44)(40,53)(43,51)(45,50)(46,52)(49,55)(54,56);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(28,39)(32,43)(34,42)(35,47)(36,49)(40,52)(50,56);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,38,29,37,41,51)(26,32,34,35,31,43,42,47)(28,36,40,50,39,49,52,56)(33,45,46,48,44,54,53,55);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,36)(29,39)(30,40)(31,44)(32,45)(34,46)(35,48)(37,49)(38,50)(41,52)(42,53)(43,54)(47,55)(51,56);(10,16)(11,18)(15,24)(17,19)(28,39)(33,44)(36,49)(40,52)(45,54)(46,53)(48,55)(50,56)Polycyclic presentation:
2^9|a^2=fhi,c^2=fgi,d^2=fi,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, 6951405);