Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23851 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1889 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)(3,7)(4,6)(5,8)(10,14)(11,12)(19,23)(22,24);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(18,20)(19,22)(23,24);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,18,21,20)(19,22,23,24);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(17,19,21,23)(18,22,20,24);(2,8)(6,7)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257312, #72257346, #72257516, … |
| maximal normal subgroup of | #72257312, #72257346, #72257516, … |
| maximal quotient of | #72308514, #72308611 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7483766);