Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1309 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,33)(28,41)(29,35)(30,36)(31,46)(32,37)(34,42)(38,48)(39,49)(40,51)(43,53)(44,52)(45,50)(47,55)(54,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,33)(28,38)(29,43)(30,45)(31,39)(32,40)(34,47)(35,53)(36,50)(37,51)(41,48)(42,55)(44,54)(46,49)(52,56);(11,17)(14,21)(18,23)(22,24)(25,28,32,29)(26,34,37,52)(27,38,40,43)(30,42,31,44)(33,47,51,56)(35,46,41,36)(39,54,45,55)(48,50,53,49);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,32,28)(26,35,37,41)(27,38,40,43)(30,42,31,44)(33,48,51,53)(34,46,52,36)(39,55,45,54)(47,50,56,49);(1,3,4,7)(2,6,5,8)(25,30,32,31)(26,36,37,46)(27,39,40,45)(28,42,29,44)(33,49,51,50)(34,35,52,41)(38,54,43,55)(47,48,56,53)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=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, 6343526);