Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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×#2537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2630 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)(8,12)(9,11)(13,14,16,20,18,15,17,19);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,19)(15,20)(16,17);(2,4)(5,8,9,7)(6,10,11,12)(13,15,16,19,18,14,17,20);(2,4)(13,14,16,20,18,15,17,19);(14,15)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282833, #72283069, #72283300, … |
| maximal normal subgroup of | #72282833, #72283069, #72283300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354384);