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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2727 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)(23,28)(24,30)(25,31)(26,40)(27,32)(29,37)(33,42)(34,43)(35,50)(36,44)(38,45)(39,46)(41,47)(48,51)(49,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,36)(22,28,32,44)(24,33,38,48)(25,34,26,35)(29,41,46,52)(30,42,45,51)(31,43,40,50)(37,47,39,49);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,33)(25,37)(26,39)(27,38)(28,41)(30,40)(31,45)(32,46)(34,47)(35,49)(36,48)(42,50)(43,51)(44,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,30)(23,33)(25,39)(26,37)(27,38)(28,42)(29,40)(31,46)(32,45)(34,49)(35,47)(36,48)(41,50)(43,52)(44,51);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,31,32,40)(23,34,36,35)(24,37,38,39)(28,43,44,50)(29,45,46,30)(33,47,48,49)(41,51,52,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,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, 6436839);