Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(9,10,15,20)(11,21,23,32)(12,26,29,33)(13,19,14,18)(16,25,31,24)(17,27,34,28)(22,36,37,30)(35,38,40,39);(2,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,25)(14,24)(15,23)(17,30)(18,32)(19,21)(20,31)(26,40)(27,39)(28,38)(29,37)(33,35)(34,36);(6,8)(9,12,15,29)(10,17,20,34)(11,22,23,37)(13,28,14,27)(16,30,31,36)(18,26,19,33)(21,35,32,40)(24,39,25,38);(6,8);(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321020 |
| maximal normal subgroup of | #72321020 |
| maximal quotient of | #72322100 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344720);