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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6222 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,25)(17,27)(22,36)(23,31)(24,32)(28,34)(29,33)(30,38)(35,39)(37,40);(2,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,25)(15,24)(17,30)(18,21)(19,32)(20,31)(26,40)(27,37)(28,39)(29,38)(33,35)(34,36);(6,8)(9,12,15,29)(10,17,20,34)(11,22,24,38)(13,27,14,28)(16,30,31,36)(18,33,19,26)(21,35,32,40)(23,37,25,39);(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,25)(12,27,29,28)(16,21,31,32)(17,33,34,26)(22,37,38,39)(30,35,36,40);(11,24)(16,31)(21,32)(22,38)(23,25)(30,36)(35,40)(37,39)Polycyclic presentation:
2^9|c^2=i,d^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 | #72314561, #72314786, #72314807, … |
| maximal normal subgroup of | #72314561, #72314786, #72314807, … |
| maximal quotient of | #72315697, #72315945, #72315956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344722);