Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,23)(20,22)(21,24);(9,10)(11,16)(12,14)(13,15)(18,22)(19,20)(23,24);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,21,20)(18,23,24,22);(10,14)(13,16)(17,19,21,20)(18,23,24,22);(1,4,5,3)(2,6,7,8)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257879, #72264340, #72281491 |
| maximal normal subgroup of | #72257879, #72264340, #72281491 |
| maximal quotient of | #72307233 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376167);