Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,34)(24,30)(25,33)(26,32)(27,31)(28,38)(29,40)(35,44)(36,48)(37,45)(39,46)(41,49)(42,47)(43,52)(50,51);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,35)(25,37)(26,38)(27,39)(29,43)(30,48)(31,45)(32,34)(33,46)(36,41)(40,51)(42,52)(44,49)(47,50);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,26,41)(22,29,32,47)(23,35,38,36)(25,40,27,42)(28,43,34,50)(30,33,49,31)(37,51,39,52)(44,45,48,46);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,30)(23,36)(25,42)(26,41)(27,40)(28,44)(29,31)(32,49)(33,47)(34,48)(35,38)(37,51)(39,52)(43,46)(45,50);(6,12)(7,14)(11,20)(13,15)(21,25,26,27)(22,31,32,33)(23,37,38,39)(24,40,41,42)(28,45,34,46)(29,30,47,49)(35,51,36,52)(43,48,50,44)Polycyclic presentation:
2^9|a^2=fhi,c^2=hi,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, 7011766);