Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,17,28,16)(7,12,9,24,19,14,26,30)(8,20,22,35,23,31,27,29)(13,25,32,18,34,21,33,36);(2,3)(5,7,10,9,11,19,28,26)(6,12,15,24,17,14,16,30)(8,21,22,36,23,25,27,18)(13,31,32,29,34,20,33,35);(5,8,11,23)(6,13,17,34)(7,18,19,36)(9,25,26,21)(10,27,28,22)(12,29,14,35)(15,33,16,32)(20,24,31,30);(1,3,4,2)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286595, #72286658, D4×#2180, … |
| maximal normal subgroup of | #72286595, #72286658, D4×#2180, … |
| maximal quotient of | #72287327, #72287516, #72309546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405531);