Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,7)(8,9)(11,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(5,7)(6,8)(9,12)(10,11)(15,21)(18,25)(22,27)(26,28);(15,21)(18,25)(22,27)(26,28);(5,8,10,9)(6,11,12,7)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)Polycyclic presentation:
2^9|a^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72304677, #72304685, #72304772, … |
| maximal normal subgroup of | #72304677, #72304685, #72304772, … |
| maximal quotient of | #72304919, #72304945, #72310233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403370);