Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1735 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)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,34)(24,30)(25,40)(26,32)(27,33)(28,31)(29,37)(35,46)(36,50)(38,52)(39,51)(41,48)(42,45)(43,47)(44,49);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,29)(24,38)(25,35)(26,39)(27,37)(28,43)(30,48)(31,46)(32,49)(33,34)(36,41)(40,47)(42,44)(45,51)(50,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,30)(23,26)(25,39)(27,36)(28,44)(29,45)(31,49)(32,34)(33,50)(35,41)(37,42)(38,43)(40,51)(46,48)(47,52);(6,12)(10,13)(11,18)(17,19)(21,25,27,28)(22,31,33,40)(23,35,37,43)(24,39,36,44)(26,41,42,38)(29,46,34,47)(30,49,50,51)(32,52,45,48);(7,15)(10,13)(14,20)(17,19)(21,26,27,42)(22,32,33,45)(23,36,37,24)(25,38,28,41)(29,30,34,50)(31,48,40,52)(35,39,43,44)(46,51,47,49)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576927);