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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6354 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)(13,17)(14,19)(15,18)(16,22)(21,24);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,18,24)(13,21,14,22)(17,20,19,23);(6,8)(9,12,15,11)(10,16,18,24)(13,22,14,21)(17,20,19,23);(6,8)(9,13,15,14)(10,17,18,19)(11,21,12,22)(16,20,24,23);(10,18)(16,24)(17,19)(20,23)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^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, #72249645, #72264307, … |
| maximal normal subgroup of | #72239231, #72249645, #72264307, … |
| maximal quotient of | #72249798, #72280282, #72281527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344713);