Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,31,36)(27,32,39,48)(28,41,29,34)(30,44,45,35)(33,43,50,42)(37,51,38,47)(40,55,54,49)(46,53,56,52);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27)(26,32)(28,37)(29,38)(30,40)(31,39)(33,46)(34,47)(35,49)(36,48)(41,51)(42,52)(43,53)(44,55)(45,54)(50,56);(1,3)(2,5)(4,8)(6,7)(25,28,31,29)(26,33,36,50)(27,37,39,38)(30,43,45,42)(32,46,48,56)(34,44,41,35)(40,53,54,52)(47,55,51,49);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,31,28)(26,34,36,41)(27,38,39,37)(30,43,45,42)(32,47,48,51)(33,44,50,35)(40,53,54,52)(46,55,56,49);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(27,39)(28,29)(32,48)(33,50)(34,41)(40,54)(42,43)(49,55)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7214374);