Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2754 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)(21,22)(23,27)(24,29)(25,28)(26,31)(30,37)(32,39)(33,38)(34,41)(35,43)(36,42)(40,47)(44,48)(45,50)(46,49)(51,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,34)(22,27,31,41)(24,32,35,45)(25,33,37,47)(28,38,30,40)(29,39,43,50)(36,46,48,52)(42,49,44,51);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,28)(25,37)(27,38)(29,42)(30,31)(33,47)(36,48)(39,49)(40,41)(43,44)(46,52)(50,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24,26,35)(22,29,31,43)(23,32,34,45)(25,36,37,48)(27,39,41,50)(28,42,30,44)(33,46,47,52)(38,49,40,51);(6,12)(10,13)(11,20)(18,19)(24,35)(29,43)(32,45)(36,48)(39,50)(42,44)(46,52)(49,51)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436840);