Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6,8,7)(13,14,17,18)(15,19,16,20);(10,12)(14,18)(19,20);(1,2,4,3)(5,6,8,7)(9,10,11,12)(13,15,17,16)(14,19,18,20);(1,3,4,2)(5,7,8,6)(13,16,17,15)(14,20,18,19);(13,14)(15,19)(16,20)(17,18)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=gi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72261336, #72261461, #72261710, … |
| maximal normal subgroup of | #72261336, #72261461, #72261710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 441621);