Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #30397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,25,30)(23,32,34,27)(24,36,38,29)(26,28,40,31)(33,42,37,45)(35,43,48,41)(39,46,50,44)(47,51,49,52);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23)(22,27)(24,37)(25,34)(26,35)(28,41)(29,45)(30,32)(31,43)(33,38)(36,42)(39,49)(40,48)(44,52)(46,51)(47,50);(2,4)(22,28)(26,40)(27,41)(29,44)(30,31)(32,43)(35,48)(36,46)(39,50)(42,51)(45,52)(47,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,42)(28,44)(30,36)(31,46)(32,45)(34,37)(35,47)(40,50)(41,51)(43,52)(48,49);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(24,38)(27,32)(29,36)(35,48)(39,50)(41,43)(44,46)Polycyclic presentation:
2^9|a^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6583019);