Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207830 |
| 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 | #28143 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,34)(27,35,38,31)(28,39,41,52)(29,33,44,51)(32,46,48,36)(37,49,42,45)(40,55,43,56)(47,54,50,53);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,36,41,46)(29,42,44,37)(32,39,48,52)(33,49,51,45)(40,54,43,53)(47,56,50,55);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,36,30,41,38,46)(26,32,31,39,34,48,35,52)(29,43,37,54,44,40,42,53)(33,50,45,56,51,47,49,55);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29)(26,33)(27,37)(28,40)(30,44)(31,45)(32,47)(34,51)(35,49)(36,53)(38,42)(39,55)(41,43)(46,54)(48,50)(52,56);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(29,44)(31,35)(32,48)(37,42)(39,52)(40,43)(53,54)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118497);