Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #52071 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26,31,36)(27,32,37,49)(28,41,45,33)(29,34,46,40)(30,35,48,39)(38,51,42,50)(43,54,47,52)(44,55,56,53);(1,3,4,7)(2,5,6,8)(10,15)(11,18)(13,21)(20,23)(27,37)(29,46)(32,49)(34,40)(38,42)(43,47)(50,51)(52,54);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,27)(26,32)(28,42)(29,39)(30,34)(31,37)(33,51)(35,46)(36,49)(38,45)(40,48)(41,50)(43,55)(44,52)(47,53)(54,56);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,31,45)(26,33,36,41)(27,38,37,42)(29,47,46,43)(30,44,48,56)(32,50,49,51)(34,54,40,52)(35,53,39,55);(1,3)(2,5)(4,7)(6,8)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,29)(26,34)(27,39)(28,43)(30,32)(31,46)(33,52)(35,37)(36,40)(38,53)(41,54)(42,55)(44,51)(45,47)(48,49)(50,56);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,30)(26,35)(27,40)(28,44)(29,32)(31,48)(33,53)(34,37)(36,39)(38,52)(41,55)(42,54)(43,51)(45,56)(46,49)(47,50)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837696);