Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,31)(23,28,34,43)(24,36,38,44)(25,30,26,39)(29,40,45,37)(32,46,48,51)(33,42,35,49)(41,50,52,47);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,32)(25,33)(26,35)(27,34)(29,41)(30,42)(31,43)(36,46)(37,47)(38,48)(39,49)(40,50)(44,51)(45,52);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,38)(22,29,31,45)(23,32,34,48)(25,37,26,40)(28,41,43,52)(30,44,39,36)(33,47,35,50)(42,51,49,46);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,31,39)(23,33,34,35)(24,37,38,40)(28,42,43,49)(29,44,45,36)(32,47,48,50)(41,51,52,46);(6,12)(10,18)(11,20)(17,19)(23,34)(28,43)(32,48)(33,35)(41,52)(42,49)(46,51)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6384840);