Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C162 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C82 | C22 |
| commutator | C2 | C162 |
| socle, prime radical | C23 | C82 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20);(2,4)(21,22,23,26,24,27,29,32,25,28,30,33,31,34,35,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc320 conjugacy classes — view the full character table.
| maximal subgroup of | #72234260 |
| maximal normal subgroup of | #72234260 |
| maximal quotient of | #72210714, #72234310, #72234311 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91);