Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,14,18,15,20,13,19)(11,21,23,16,25,32,24,31)(12,26,28,17,29,34,27,35)(22,30,37,36,39,33,38,40);(5,6,7,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,33)(18,32)(19,21)(20,31)(26,36)(27,38)(28,37)(29,39)(30,35)(34,40);(2,4)(9,12,15,29)(10,17,20,35)(11,22,25,39)(13,28,14,27)(16,30,31,33)(18,26,19,34)(21,36,32,40)(23,38,24,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239913, #72314513, #72314520, … |
| maximal normal subgroup of | #72239913, #72314513, #72314520, … |
| maximal quotient of | #72239875, #72239876, #72239877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30802);