Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6)(7,8)(10,14)(15,16)(18,21)(19,23)(22,24);(2,3)(9,10)(11,15)(12,16)(13,14)(18,22)(21,24);(1,2,4,3)(9,11,13,12)(10,15,14,16)(17,18,19,21,20,22,23,24);(1,3,4,2)(6,8)(9,12,13,11)(10,15,14,16)(18,22)(21,24)Polycyclic presentation:
2^9|a^2=gi,c^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239733, #72239765, #72239776 |
| maximal normal subgroup of | #72239733, #72239765, #72239776 |
| maximal quotient of | #72257365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341576);