Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,30)(23,32,35,27)(24,37,38,29)(26,41,40,31)(28,42,46,39)(33,47,34,44)(36,45,52,49)(43,50,48,51);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,35)(22,27,30,32)(24,33,38,34)(26,36,40,52)(28,43,46,48)(29,47,37,44)(31,45,41,49)(39,50,42,51);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(23,33)(25,38)(26,42)(27,43)(29,31)(30,46)(32,48)(34,35)(36,51)(37,41)(39,40)(44,49)(45,47)(50,52);(6,12)(10,13)(11,18)(17,19)(21,24)(22,29)(23,34)(25,38)(26,39)(27,44)(28,31)(30,37)(32,47)(33,35)(36,51)(40,42)(41,46)(43,49)(45,48)(50,52);(2,4)(7,15)(10,13)(14,20)(17,19)(23,35)(24,38)(27,32)(28,46)(29,37)(36,52)(39,42)(45,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^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,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577236);