Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,37)(27,38,41,33)(28,35,29,45)(30,36,32,47)(34,46,52,44)(39,49,43,53)(40,54,42,50)(48,55,51,56);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27)(26,33)(28,43)(29,39)(30,40)(31,41)(32,42)(34,51)(35,49)(36,50)(37,38)(44,56)(45,53)(46,55)(47,54)(48,52);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,31,29)(26,34,37,52)(27,39,41,43)(30,44,32,46)(33,48,38,51)(35,36,45,47)(40,55,42,56)(49,54,53,50);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(25,29,31,28)(26,35,37,45)(27,39,41,43)(30,44,32,46)(33,49,38,53)(34,36,52,47)(40,56,42,55)(48,54,51,50);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,31,32)(26,36,37,47)(27,40,41,42)(28,44,29,46)(33,50,38,54)(34,45,52,35)(39,55,43,56)(48,49,51,53)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538621);