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 | C2×#538 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(15,18)(16,21)(22,24)(25,26,31,32)(27,33,40,47)(28,34,43,48)(29,35,44,39)(30,36,46,38)(37,49,53,55)(41,50,45,52)(42,51,56,54);(1,2,4,6)(3,5,8,7)(12,20)(16,21)(17,23)(22,24)(26,32)(29,44)(33,47)(34,48)(36,38)(41,45)(49,55)(51,54);(1,2)(3,5)(4,6)(7,8)(9,10,14,13)(11,15,19,18)(12,16,20,21)(17,22,23,24)(25,27)(26,33)(28,37)(29,38)(30,35)(31,40)(32,47)(34,49)(36,44)(39,46)(41,51)(42,52)(43,53)(45,54)(48,55)(50,56);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,13,18)(12,17,20,23)(16,22,21,24)(25,28,31,43)(26,34,32,48)(27,37,40,53)(29,45,44,41)(30,42,46,56)(33,49,47,55)(35,52,39,50)(36,51,38,54);(9,12)(10,16)(11,17)(13,21)(14,20)(15,22)(18,24)(19,23)(25,29)(26,35)(27,38)(28,41)(30,33)(31,44)(32,39)(34,50)(36,40)(37,51)(42,55)(43,45)(46,47)(48,52)(49,56)(53,54);(9,13,14,10)(11,18,19,15)(12,21,20,16)(17,24,23,22)(25,30)(26,36)(27,39)(28,42)(29,33)(31,46)(32,38)(34,51)(35,40)(37,50)(41,55)(43,56)(44,47)(45,49)(48,54)(52,53)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833851);