Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,31)(27,35,38,50)(28,33,39,45)(29,42,43,34)(32,46,44,37)(36,51,52,56)(40,49,41,48)(47,53,55,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(26,31)(28,39)(32,44)(34,42)(35,50)(36,52)(40,41)(53,54);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,37,30,38,43,46)(26,32,34,35,31,44,42,50)(28,36,41,54,39,52,40,53)(33,47,49,51,45,55,48,56);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28)(26,33)(27,36)(29,41)(30,39)(31,45)(32,47)(34,49)(35,51)(37,54)(38,52)(40,43)(42,48)(44,55)(46,53)(50,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,43)(26,34,31,42)(27,37,38,46)(28,40,39,41)(32,35,44,50)(33,48,45,49)(36,53,52,54)(47,56,55,51)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7246030);