Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #34736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,20)(12,23)(13,19)(14,18)(15,17)(16,21)(22,24);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,17,24)(13,22,14,21)(18,23,19,20);(6,8)(9,12,15,11)(10,16,17,24)(13,21,14,22)(18,23,19,20);(6,8);(10,17)(16,24)(18,19)(20,23)Polycyclic presentation:
2^9|b^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239231, #72249625, #72264307, … |
| maximal normal subgroup of | #72239231, #72249625, #72264307, … |
| maximal quotient of | #72249804, #72264647, #72280154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344709);