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)(27,37,44,46,38,32,40,48)(28,41,45,50,42,34,43,49)(39,47,54,56,52,51,53,55);(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)(26,32)(28,39)(29,38)(30,40)(31,44)(33,48)(34,47)(35,37)(36,46)(41,51)(42,52)(43,53)(45,54)(49,56)(50,55);(1,3,5,7)(2,6,4,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,38)(28,42)(30,31)(32,46)(34,50)(35,36)(37,48)(41,49)(47,56)(51,55)(53,54);(3,7)(6,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,39)(29,42)(30,43)(31,45)(32,47)(33,49)(35,41)(36,50)(37,51)(38,52)(40,53)(44,54)(46,55)(48,56);(1,4,5,2)(3,8,7,6)(10,16)(14,17)(15,22)(21,23)(28,42)(34,41)(39,52)(43,45)(47,51)(49,50)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 7394589);