Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180869 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,h^a=hi,c^b=ce,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72211326, #72219659, #72219661, … |
| maximal normal subgroup of | #72211326, #72219659, #72219661, … |
| maximal quotient of | #72210893, #72257483, #72264187 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263171);