Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,32,38)(27,33,43,51)(28,44,29,35)(30,37,31,36)(34,46,52,45)(39,53,40,48)(41,50,42,49)(47,55,56,54);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,32,43)(26,33,38,51)(28,39,29,40)(30,41,31,42)(34,47,52,56)(35,48,44,53)(36,49,37,50)(45,54,46,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,34)(27,39)(29,32)(30,45)(31,46)(33,47)(35,37)(36,44)(38,52)(40,43)(41,54)(42,55)(48,50)(49,53)(51,56);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,29,32,28)(26,35,38,44)(27,40,43,39)(30,45,31,46)(33,48,51,53)(34,36,52,37)(41,54,42,55)(47,49,56,50);(10,16)(11,18)(15,24)(17,19)(25,30,32,31)(26,36,38,37)(27,41,43,42)(28,45,29,46)(33,49,51,50)(34,44,52,35)(39,54,40,55)(47,53,56,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6942474);