Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1431 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)(27,36)(28,39)(29,34)(30,35)(31,32)(33,40)(37,48)(38,49)(41,52)(42,45)(43,46)(44,50)(47,53)(51,56)(54,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,43,44,38)(33,39,48,45)(34,49,50,46)(41,51,54,53)(47,55,56,52);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,40,30,42,27,37)(26,33,36,39,35,48,32,45)(29,41,43,53,44,54,38,51)(34,47,49,52,50,56,46,55);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,38)(28,41)(30,44)(31,43)(32,46)(33,47)(35,50)(36,49)(37,51)(39,52)(40,53)(42,54)(45,55)(48,56);(10,16)(11,18)(15,24)(17,19)(26,35)(28,42)(29,44)(32,36)(37,40)(38,43)(47,56)(52,55)Polycyclic presentation:
2^9|a^2=fh,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6948731);