Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,36)(27,32,37,47)(28,34,41,52)(29,35,30,33)(38,46,40,50)(39,48,43,45)(42,49,44,51)(53,55,54,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,40)(29,43)(30,39)(31,37)(33,48)(34,50)(35,45)(36,47)(38,41)(42,53)(44,54)(46,52)(49,55)(51,56);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,33)(27,37)(28,41)(29,30)(32,45)(34,51)(35,36)(46,56)(47,48)(49,52)(50,55)(53,54);(11,18)(14,22)(17,19)(21,23)(25,28)(26,34)(27,38)(29,44)(30,42)(31,41)(32,46)(33,49)(35,51)(36,52)(37,40)(39,54)(43,53)(45,55)(47,50)(48,56);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(25,29,31,30)(26,33,36,35)(27,39,37,43)(28,42,41,44)(32,45,47,48)(34,51,52,49)(38,53,40,54)(46,56,50,55)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6386708);