Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(8,13)(9,10)(14,29)(15,26)(16,33)(17,34)(18,28)(19,24)(21,22)(23,27)(25,31)(30,35)(32,36);(5,6,13,27,10,16,22,26,11,15,21,33,9,23,8,12)(7,17,28,25,19,14,30,36,20,32,35,29,24,31,18,34);(5,7)(6,14)(8,18)(9,24)(10,19)(11,20)(12,25)(13,28)(15,31)(16,32)(17,23)(21,35)(22,30)(26,29)(27,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320253, #72320338, #72320481, … |
| maximal normal subgroup of | #72320253, #72320338, #72320481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55532);