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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,36)(27,38,42,32)(28,34,45,53)(29,37,31,35)(33,46,51,44)(39,48,40,52)(41,49,43,50)(47,55,54,56);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,39)(29,41)(30,42)(31,43)(33,47)(34,52)(35,49)(36,38)(37,50)(40,45)(44,55)(46,56)(48,53)(51,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,45)(26,33,36,51)(27,39,42,40)(29,44,31,46)(32,47,38,54)(34,37,53,35)(41,55,43,56)(48,49,52,50);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,40)(29,46)(30,45)(31,44)(32,48)(33,35)(36,53)(37,51)(38,52)(39,42)(41,55)(43,56)(47,50)(49,54);(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,36,37)(27,41,42,43)(28,44,45,46)(32,49,38,50)(33,34,51,53)(39,55,40,56)(47,52,54,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=i,d^2=fi,e^2=i,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, 6942478);