Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1538 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,32)(28,41)(29,35)(30,34)(31,36)(33,42)(37,51)(38,50)(39,46)(40,47)(43,48)(44,49)(45,53)(52,56)(54,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,37)(29,39)(30,38)(31,40)(33,45)(34,50)(35,46)(36,47)(41,51)(42,53)(43,54)(44,52)(48,55)(49,56);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,45)(34,41)(35,48)(36,49)(38,52)(39,53)(40,54)(46,55)(47,56)(50,51);(12,20)(15,24)(17,19)(21,23)(25,29,31,30)(26,34,36,35)(27,38,40,39)(28,42,43,44)(32,46,47,50)(33,41,49,48)(37,52,54,53)(45,55,56,51);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,35,36,34)(27,39,40,38)(28,42,43,44)(32,46,47,50)(33,48,49,41)(37,53,54,52)(45,55,56,51)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6416188);