Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1793 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)(21,22,27,32)(23,28,35,44)(24,37,40,48)(25,31,26,30)(29,41,47,39)(33,49,38,52)(34,45,36,43)(42,51,46,50);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23)(22,28)(24,38)(25,34)(26,36)(27,35)(29,46)(30,43)(31,45)(32,44)(33,40)(37,52)(39,51)(41,50)(42,47)(48,49);(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,41)(27,40)(28,42)(30,37)(31,48)(32,47)(34,50)(35,38)(36,51)(43,49)(44,46)(45,52);(6,11)(10,16)(12,18)(17,20)(21,25,27,26)(22,30,32,31)(23,34,35,36)(24,39,40,41)(28,43,44,45)(29,37,47,48)(33,50,38,51)(42,49,46,52);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,35)(28,44)(33,38)(34,36)(42,46)(43,45)(49,52)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343891);