Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#496 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,7)(13,14,18,15)(16,19,20,17);(5,6,8,7)(9,10)(11,12)(13,15,17,16,18,14,19,20);(2,4)(5,7,8,6)(10,12)(13,14,17,20,18,15,19,16);(13,16,17,14,18,20,19,15)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265180, #72265237, #72265286, … |
| maximal normal subgroup of | #72265180, #72265237, #72265286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319364);