Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26)(27,31)(28,33)(29,32)(30,34)(35,45)(36,44)(37,46)(38,43)(39,49)(40,47)(41,50)(42,48)(51,55)(52,54)(53,56);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,29,37)(26,31,32,46)(28,39,40,36)(30,35,42,38)(33,49,47,44)(34,45,48,43)(41,52,53,51)(50,54,56,55);(1,4,5,3)(2,6,7,8)(26,32)(27,35)(28,40)(30,42)(31,43)(36,52)(37,38)(39,51)(44,55)(45,46)(49,54)(50,56);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,40)(26,33,32,47)(27,36,37,39)(30,41,42,53)(31,44,46,49)(34,50,48,56)(35,51,38,52)(43,54,45,55);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,35)(30,42)(31,45)(34,48)(36,51)(37,38)(39,52)(41,53)(43,46)(44,55)(49,54)(50,56)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6601502);