Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #42796 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,39)(29,34)(30,35)(31,32)(33,37)(38,48)(40,52)(41,50)(42,45)(43,49)(44,47)(46,51)(53,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,31)(26,32,35,36)(28,37,41,44)(29,42,43,38)(33,39,47,50)(34,48,49,45)(40,53,54,51)(46,55,56,52);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,41)(26,33,35,47)(27,37,31,44)(29,40,43,54)(32,39,36,50)(34,46,49,56)(38,51,42,53)(45,52,48,55);(3,8)(5,7)(11,18)(12,20)(14,22)(15,24)(25,29,30,43)(26,34,35,49)(27,38,31,42)(28,40,41,54)(32,45,36,48)(33,46,47,56)(37,51,44,53)(39,52,50,55);(2,6)(5,7)(11,18)(14,22)(17,19)(21,23)(26,35)(29,43)(32,36)(33,47)(38,42)(39,50)(40,54)(51,53)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7247258);