Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,7)(5,6)(10,14)(12,20)(15,18)(19,23)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39)(41,42)(43,47)(44,46)(45,52)(48,53)(49,55)(50,51)(54,56);(1,3,6,4)(2,7,5,8)(10,14)(11,17)(16,21)(19,23)(28,35)(31,33)(36,40)(38,39)(42,44)(45,53)(47,51)(49,56);(3,4)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38)(41,43)(42,47)(44,51)(45,49)(46,50)(48,54)(52,55)(53,56);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,44,46,42)(43,47,50,51)(45,52,53,48)(49,54,56,55);(1,5,6,2)(3,7,4,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,45)(42,48)(43,49)(44,52)(46,53)(47,54)(50,56)(51,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=gh,c^2=fghi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038820);