Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| 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 | |
3388 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)(13,14,16,20,18,22,17,21)(15,23,26,19,25,28,24,27);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(6,9)(7,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28);(6,9)(7,11)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | #72257329, #72257349, #72257464, … |
| maximal normal subgroup of | #72257329, #72257349, #72257464, … |
| maximal quotient of | #72265079, #72305572, #72305627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503440);