Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,11,7,10,15)(8,16,17,14,19,20,18,12);(1,2)(3,4)(5,7)(6,11)(8,16,19,20)(9,13)(10,15)(12,18,14,17);(2,4)(5,8,9,18,11,19,10,17)(6,12,13,20,7,14,15,16);(8,17)(9,10)(12,16)(13,15)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284694, #72285552, #72285592, … |
| maximal normal subgroup of | #72284694, #72285552, #72285592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364082);