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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1244 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)(27,36)(28,33)(29,34)(30,32)(31,38)(35,43)(37,48)(39,46)(40,45)(41,50)(42,47)(44,52)(49,54)(51,56)(53,55);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,38)(26,31,34,36)(28,40,41,37)(30,39,43,52)(32,44,35,46)(33,48,50,45)(42,53,54,51)(47,56,49,55);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(26,32)(28,41)(30,43)(31,44)(33,49)(34,35)(36,46)(37,40)(39,52)(45,56)(47,50)(48,55);(1,4,5,3)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,47)(34,50)(35,49)(36,48)(38,40)(39,51)(43,54)(44,55)(46,56)(52,53);(10,16)(11,18)(15,24)(17,19)(26,34)(28,41)(31,36)(32,35)(37,40)(42,54)(44,46)(51,53)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,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, 6943005);