Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #42685 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,31)(28,33)(29,32)(30,34)(35,41)(36,43)(37,42)(38,44)(39,48)(40,46)(45,51)(47,52)(49,55)(50,53)(54,56);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,34,44)(28,36,39,49)(29,37,41,51)(32,42,35,45)(33,43,48,55)(40,50,52,56)(46,53,47,54);(1,2,5,4)(3,6,8,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(28,39)(29,41)(31,42)(33,47)(34,35)(36,49)(37,51)(43,54)(44,45)(46,48)(53,55);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,39)(26,33,34,48)(27,36,38,49)(29,40,41,52)(31,43,44,55)(32,46,35,47)(37,50,51,56)(42,53,45,54);(10,16)(11,18)(15,24)(17,19)(26,34)(28,39)(31,44)(32,35)(36,49)(40,52)(42,45)(50,56)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fh,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7012777);