Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20,18,15,17,19);(1,2)(3,4)(14,19)(15,20)(16,17);(2,4)(6,9)(10,12)(13,15,16,19,18,14,17,20);(2,4)(13,14,16,20,18,15,17,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72343941, #72343949, #72343964 |
| maximal normal subgroup of | #72343941, #72343949, #72343964 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363649);