Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36)(27,32,39,49)(28,40,29,34)(30,44,45,35)(33,43,51,42)(37,47,41,56)(38,53,55,48)(46,54,50,52);(2,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,41)(29,37)(30,38)(31,39)(33,50)(34,47)(35,48)(36,49)(40,56)(42,54)(43,52)(44,53)(45,55)(46,51);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,33)(27,37)(29,31)(30,43)(32,46)(34,35)(36,51)(38,54)(39,41)(40,44)(42,45)(47,53)(48,56)(49,50)(52,55);(1,4,5,3)(2,8,6,7)(12,20)(15,24)(17,19)(21,23)(25,29,31,28)(26,34,36,40)(27,37,39,41)(30,43,45,42)(32,47,49,56)(33,44,51,35)(38,52,55,54)(46,48,50,53);(11,18)(14,22)(17,19)(21,23)(28,29)(33,51)(34,40)(37,41)(42,43)(46,50)(47,56)(52,54)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6416958);