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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2802 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)(21,22,26,30)(23,31,32,44)(24,28,35,42)(25,37,39,29)(27,38,40,34)(33,45,46,52)(36,49,48,43)(41,50,51,47);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,33)(25,38)(26,32)(28,41)(29,44)(30,40)(31,37)(34,39)(35,46)(36,50)(42,51)(43,52)(45,49)(47,48);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(23,32)(27,40)(31,44)(33,46)(34,38)(41,51)(45,52)(47,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,28)(23,33)(25,36)(26,35)(27,41)(29,43)(30,42)(31,45)(32,46)(34,47)(37,49)(38,50)(39,48)(40,51)(44,52);(6,12)(10,13)(11,20)(18,19)(23,32)(24,35)(27,40)(28,42)(31,44)(34,38)(36,48)(43,49)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436207);