Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,11,17,8,14)(7,12,21,27,18,28,19,26)(9,15,22,31,23,13,24,30)(20,25,33,35,32,29,34,36);(1,2,3,4)(5,7)(6,12)(8,19)(9,20)(10,21)(11,18)(13,29)(14,26)(15,25)(16,27)(17,28)(22,33)(23,32)(24,34)(30,36)(31,35);(6,13)(7,18)(9,23)(12,25)(14,30)(15,17)(16,31)(19,21)(22,24)(26,35)(27,36)(28,29);(5,8,11,10)(6,14,17,16)(7,19,18,21)(9,22,23,24)(12,26,28,27)(13,30,15,31)(20,33,32,34)(25,35,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72293175, #72293177, #72293180, … |
| maximal normal subgroup of | #72293175, #72293177, #72293180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400919);