Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61030 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,15,12,16,10,28)(7,17,21,13,18,33,29,31)(8,14,23,9,24,26,30,27)(19,35,32,20,36,34,22,25);(5,7)(6,13)(8,22)(9,25)(10,29)(11,21)(12,18)(14,34)(15,33)(16,31)(17,28)(19,23)(20,27)(24,32)(26,35)(30,36);(2,4)(7,18)(13,31)(17,33)(19,36)(20,25)(21,29)(22,32)(34,35);(5,8)(6,14)(7,19)(9,15)(10,30)(11,23)(12,24)(13,20)(16,26)(17,35)(18,36)(21,32)(22,29)(25,31)(27,28)(33,34);(5,9,11,26,12,27,10,14)(6,8,15,23,16,24,28,30)(7,20,21,34,18,25,29,35)(13,32,33,36,31,22,17,19)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72343392, #72343646, #72343656 |
| maximal normal subgroup of | #72343392, #72343646, #72343656 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291210);