Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499780 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(7,8)(11,12)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(1,2,3,4)(6,11)(10,12)(15,21)(16,23)(18,24)(19,26);(5,7,9,8)(6,10,11,12)(13,16,17,23)(14,19,20,26)(15,22,21,27)(18,25,24,28);(6,11)(10,12)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72304376, #72304434, #72304443, … |
| maximal normal subgroup of | #72304376, #72304434, #72304443, … |
| maximal quotient of | #72305039, #72305781, #72306057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6433917);