Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | |
1246 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)(25,26,31,33,29,35,30,36)(27,37,43,45,38,32,40,47)(28,34,44,48,41,50,42,49)(39,51,54,55,52,46,53,56);(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)(26,32)(28,39)(29,38)(30,40)(31,43)(33,47)(34,46)(35,37)(36,45)(41,52)(42,53)(44,54)(48,56)(49,55)(50,51);(1,3,5,7)(2,6,4,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,38)(28,41)(30,31)(32,45)(34,49)(35,36)(37,47)(46,56)(48,50)(51,55)(53,54);(3,7)(6,8)(11,19)(14,17)(18,24)(21,23)(25,28,29,41)(26,34,35,50)(27,39,38,52)(30,42,31,44)(32,46,37,51)(33,48,36,49)(40,53,43,54)(45,55,47,56);(1,4,5,2)(3,8,7,6)(10,16)(14,17)(15,22)(21,23)(28,41)(34,50)(39,52)(42,44)(46,51)(48,49)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7394335);