Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,40)(38,50)(39,51)(41,46)(42,49)(43,48)(44,45)(47,54)(52,56)(53,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,38)(28,41)(35,36)(40,44)(42,43)(47,55)(51,56);(1,2,5,4)(3,6,8,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27,30,38)(26,33,32,45)(28,39,41,52)(29,40,31,44)(34,47,46,55)(35,37,36,50)(42,53,43,54)(48,56,49,51);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,39)(29,43)(30,41)(31,42)(32,46)(33,47)(35,49)(36,48)(37,51)(38,52)(40,54)(44,53)(45,55)(50,56);(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,32,36)(27,40,38,44)(28,42,41,43)(33,37,45,50)(34,48,46,49)(39,53,52,54)(47,56,55,51)Polycyclic presentation:
2^9|a^2=fh,c^2=fhi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7012806);