Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,15,18)(11,13,17,14)(12,20,21,19)(16,22,24,23);(1,2,3,4)(5,6)(7,8)(9,11,14,10,15,17,13,18)(12,19,23,16,21,20,22,24);(6,8)(9,12,14,23,15,21,13,22)(10,16,17,20,18,24,11,19);(6,8)(12,21)(16,24)(19,20)(22,23);(9,13,15,14)(10,17,18,11)(12,22,21,23)(16,20,24,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^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,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224791, #72224933, #72224936, … |
| maximal normal subgroup of | #72224791, #72224933, #72224936, … |
| maximal quotient of | #72224774, #72264804, #72310612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350432);