Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,29,34)(27,36,38,31)(28,40,41,33)(30,44,43,35)(32,45,49,42)(37,47,53,51)(39,48,55,52)(46,54,50,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,29,38)(26,31,34,36)(28,37,41,53)(30,39,43,55)(32,46,49,50)(33,47,40,51)(35,48,44,52)(42,54,45,56);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,37)(29,41)(30,45)(31,46)(33,35)(34,49)(36,50)(38,53)(39,56)(40,44)(42,43)(47,48)(51,52)(54,55);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,47)(32,35)(34,40)(36,51)(38,53)(39,54)(43,45)(44,49)(46,48)(50,52)(55,56);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,38)(28,41)(31,36)(32,49)(33,40)(39,55)(42,45)(48,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6685045);