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 | #42664 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1273 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)(25,26,31,33,30,36,29,35)(27,32,42,44,38,46,39,47)(28,34,43,48,40,49,41,50)(37,45,53,54,52,56,51,55);(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)(28,37)(29,39)(30,38)(31,42)(33,44)(34,45)(35,47)(36,46)(40,52)(41,51)(43,53)(48,54)(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)(29,31)(32,44)(34,48)(35,36)(39,42)(41,43)(45,54)(46,47)(49,50)(51,53)(55,56);(3,7)(6,8)(11,19)(14,17)(18,24)(21,23)(25,28,30,40)(26,34,36,49)(27,37,38,52)(29,41,31,43)(32,45,46,56)(33,48,35,50)(39,51,42,53)(44,54,47,55);(1,4,5,2)(3,8,7,6)(10,16)(14,17)(15,22)(21,23)(27,38)(28,40)(32,46)(34,49)(39,42)(41,43)(44,47)(48,50)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=ehi,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, 7394218);