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 | #56811 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4081 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)(13,21)(14,15)(16,22)(19,20)(23,24)(26,31)(27,35)(29,41)(33,44)(36,38)(39,53)(46,51)(55,56);(3,7)(5,8)(10,15)(11,18)(16,21)(19,24)(26,31)(32,43)(33,44)(34,42)(45,47)(46,51)(48,54)(55,56);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,23)(22,24)(25,26)(27,32)(28,33)(29,42)(30,31)(34,41)(35,43)(36,45)(37,51)(38,47)(39,54)(40,44)(46,49)(48,53)(50,56)(52,55);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,27,30,35)(26,32,31,43)(28,38,40,36)(29,37,41,49)(33,47,44,45)(34,46,42,51)(39,52,53,50)(48,56,54,55);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28,30,40)(26,33,31,44)(27,36,35,38)(29,39,41,53)(32,45,43,47)(34,48,42,54)(37,50,49,52)(46,55,51,56);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,29)(26,34)(27,37)(28,39)(30,41)(31,42)(32,46)(33,48)(35,49)(36,50)(38,52)(40,53)(43,51)(44,54)(45,55)(47,56)Polycyclic presentation:
2^9|a^2=hi,d^2=gi,e^2=gi,f^2=g,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, 8833975);