Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | |
1212 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)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,31)(28,39)(29,33)(30,34)(32,41)(35,53)(36,51)(37,47)(38,46)(40,56)(42,52)(43,50)(44,49)(45,54)(48,55);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,29,37)(26,31,33,47)(28,40,41,55)(30,36,43,38)(32,48,39,56)(34,51,50,46)(35,44,54,42)(45,52,53,49);(2,7)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,32)(27,35)(29,41)(30,44)(31,45)(33,39)(34,52)(36,40)(37,54)(38,55)(42,43)(46,56)(47,53)(48,51)(49,50);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(28,41)(32,39)(35,54)(40,55)(42,44)(45,53)(48,56)(49,52);(1,4,5,3)(2,8,7,6)(11,21)(15,18)(19,20)(23,24)(26,33)(27,36)(30,43)(31,46)(32,39)(35,40)(37,38)(42,44)(45,56)(47,51)(48,53)(54,55)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=f,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6626963);