Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1433 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)(28,39)(29,33)(30,34)(32,36)(35,38)(37,46)(40,48)(41,55)(42,45)(43,56)(44,51)(47,52)(49,53)(50,54);(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)(26,31,34,35)(28,40,42,36)(29,37,44,54)(32,39,48,45)(33,46,51,50)(41,53,43,52)(47,55,49,56);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,38,40,30,42,27,36)(26,32,31,45,34,48,35,39)(29,43,54,52,44,41,37,53)(33,49,46,55,51,47,50,56);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,29)(26,33)(27,37)(28,41)(30,44)(31,46)(32,47)(34,51)(35,50)(36,52)(38,54)(39,55)(40,53)(42,43)(45,56)(48,49);(10,16)(11,18)(15,24)(17,19)(26,34)(29,44)(31,35)(32,48)(37,54)(39,45)(41,43)(52,53)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fgi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6948427);