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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1221 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)(3,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,29,33)(27,31,37,47)(28,39,41,32)(30,34,43,51)(35,53,54,45)(36,46,38,48)(40,56,55,49)(42,52,44,50);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,31)(28,40)(29,37)(30,36)(32,49)(33,47)(34,46)(35,42)(38,43)(39,56)(41,55)(44,54)(45,50)(48,51)(52,53);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,29,41)(26,32,33,39)(27,35,37,54)(30,44,43,42)(31,45,47,53)(34,52,51,50)(36,40,38,55)(46,49,48,56);(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)(49,56)(50,52);(1,4,5,3)(2,7,8,6)(11,21)(15,18)(19,20)(23,24)(27,36)(30,43)(31,46)(34,51)(35,40)(37,38)(42,44)(45,49)(47,48)(50,52)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fg,d^2=f,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, 6635079);