Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | |
1393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,4)(3,7)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,30,34)(27,35,37,51)(28,32,40,48)(29,42,43,33)(31,44,46,38)(36,50,39,54)(41,56,52,49)(45,55,47,53);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24);(1,3)(2,6)(4,8)(5,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,29,38,30,37,43,44)(26,31,33,35,34,46,42,51)(28,39,41,55,40,36,52,53)(32,47,49,54,48,45,56,50);(3,7)(6,8)(11,20)(15,18)(19,21)(23,24)(25,28)(26,32)(27,36)(29,41)(30,40)(31,45)(33,49)(34,48)(35,50)(37,39)(38,53)(42,56)(43,52)(44,55)(46,47)(51,54);(1,4,5,2)(3,8,7,6)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,37)(28,40)(31,46)(32,48)(35,51)(38,44)(41,52)(49,56)Polycyclic presentation:
2^9|a^2=fhi,c^2=fg,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=eh,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7262668);