Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2424 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2313 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)(5,6)(7,10)(8,9)(11,12)(13,14,17,20,18,21,26,27)(15,19,23,16,22,25,28,24);(6,9)(10,12)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(2,4);(13,16,17,25,18,24,26,19)(14,15,20,23,21,22,27,28);(5,7,8,11)(6,10,9,12)(15,22)(16,24)(19,25)(23,28)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^b=ei,e^d=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72239271, #72256930, #72256943, … |
| maximal normal subgroup of | #72239271, #72256930, #72256943, … |
| maximal quotient of | #72239392, #72239393, #72264277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280572);