Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,31)(28,39)(29,33)(30,35)(32,36)(34,38)(37,44)(40,52)(41,43)(42,47)(45,49)(46,50)(48,51)(53,54)(55,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,35,34)(28,36,41,50)(29,37,42,51)(32,43,46,39)(33,44,47,48)(40,49,53,56)(45,54,55,52);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,38,50,30,41,27,36)(26,32,31,43,35,46,34,39)(29,40,51,56,42,53,37,49)(33,45,44,54,47,55,48,52);(2,6)(7,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,33)(27,37)(28,40)(30,42)(31,44)(32,45)(34,48)(35,47)(36,49)(38,51)(39,52)(41,53)(43,54)(46,55)(50,56);(1,4,5,3)(2,8,6,7)(10,16)(11,18)(15,24)(17,19)(29,42)(33,47)(37,51)(40,53)(44,48)(45,55)(49,56)(52,54)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fgi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6954979);