Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,32)(27,35,39,31)(28,40,41,33)(30,34,43,46)(36,44,38,48)(37,45,52,49)(42,53,54,47)(50,56,51,55);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,39)(26,31,32,35)(28,37,41,52)(30,36,43,38)(33,45,40,49)(34,48,46,44)(42,50,54,51)(47,56,53,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(27,36)(28,41)(30,43)(31,44)(35,48)(37,51)(38,39)(45,56)(47,53)(49,55)(50,52);(10,16)(14,17)(15,22)(21,23)(25,28,29,41)(26,33,32,40)(27,37,39,52)(30,42,43,54)(31,45,35,49)(34,47,46,53)(36,50,38,51)(44,55,48,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(27,38)(30,43)(31,44)(34,46)(35,48)(36,39)(37,51)(42,54)(45,55)(47,53)(49,56)(50,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6629235);