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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1377 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,29,34)(27,36,37,31)(28,40,42,33)(30,32,44,35)(38,46,41,49)(39,45,51,47)(43,50,54,48)(52,56,53,55);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,31)(28,41)(29,37)(30,39)(32,47)(33,49)(34,36)(35,45)(38,42)(40,46)(43,53)(44,51)(48,55)(50,56)(52,54);(11,17)(14,20)(18,23)(21,24)(26,32)(27,37)(30,44)(31,45)(33,48)(34,35)(36,47)(38,41)(40,50)(43,54)(46,55)(49,56);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,28)(26,33)(27,38)(29,42)(30,43)(31,46)(32,48)(34,40)(35,50)(36,49)(37,41)(39,52)(44,54)(45,55)(47,56)(51,53);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(28,42)(33,40)(38,41)(43,54)(46,49)(48,50)(52,53)(55,56)Polycyclic presentation:
2^9|a^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,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, 6344319);