Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1638 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)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,28,32)(23,35,39,48)(24,40,42,46)(25,31,41,49)(26,34,27,33)(29,38,47,37)(30,36,50,43)(44,52,45,51);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,38,28,39,26,37)(22,29,34,48,32,47,33,35)(24,41,43,44,42,25,36,45)(30,49,40,51,50,31,46,52);(7,13)(10,17)(14,15)(18,19)(21,24,26,36,28,42,27,43)(22,30,33,46,32,50,34,40)(23,25,37,44,39,41,38,45)(29,31,35,51,47,49,48,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,25)(22,31)(23,36)(24,38)(26,44)(27,45)(28,41)(29,46)(30,48)(32,49)(33,51)(34,52)(35,50)(37,42)(39,43)(40,47);(6,12)(10,17)(11,20)(18,19)(22,32)(24,42)(25,41)(29,47)(33,34)(35,48)(36,43)(44,45)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391910);