Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,26,30)(23,31,32,44)(24,35,36,28)(25,38,40,29)(27,39,41,34)(33,47,46,45)(37,43,51,50)(42,48,49,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(23,32)(27,41)(31,44)(33,46)(34,39)(42,49)(45,47)(48,52);(7,13)(10,16)(14,19)(17,20)(21,23,26,32)(22,27,30,41)(24,33,36,46)(25,39,40,34)(28,42,35,49)(29,44,38,31)(37,52,51,48)(43,47,50,45);(6,11)(10,16)(12,18)(17,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,42)(29,43)(30,35)(31,45)(32,46)(34,48)(38,50)(39,52)(40,51)(41,49)(44,47);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(24,36)(28,35)(33,46)(37,51)(42,49)(43,50)(45,47)(48,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,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, 6343918);