Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | |
1253 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,35,29,34,44,32)(27,36,39,46,38,31,45,48)(28,40,43,49,42,33,50,52)(37,47,53,56,41,51,54,55);(1,2)(3,8)(4,5)(6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,45)(32,46,35,48)(33,51,40,47)(43,54,50,53)(49,56,52,55);(1,3,5,7)(2,6,4,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,44)(31,46)(33,49)(34,35)(36,48)(39,45)(40,52)(43,50)(47,55)(51,56)(53,54);(3,7)(6,8)(11,19)(14,17)(18,24)(21,23)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,50)(31,47,36,51)(32,49,35,52)(39,53,45,54)(46,55,48,56);(1,4,5,2)(3,8,7,6)(10,16)(14,17)(15,22)(21,23)(26,34)(28,42)(31,36)(32,35)(37,41)(43,50)(46,48)(53,54)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7394398);