Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1862 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)(5,6,10,16,11,12,9,15)(7,17,21,13,20,27,18,26)(8,22,24,29,25,14,23,30)(19,31,32,35,33,36,34,28);(1,3,4,2)(6,12)(7,18)(9,10)(13,27)(14,22)(17,26)(19,32)(20,21)(23,24)(28,36)(31,35)(33,34);(5,7)(6,13)(8,19)(9,21)(10,18)(11,20)(12,26)(14,28)(15,27)(16,17)(22,35)(23,32)(24,34)(25,33)(29,31)(30,36);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,22)(13,28)(15,30)(16,29)(17,31)(18,32)(20,33)(21,34)(26,35)(27,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312207, #72312219, #72312228, … |
| maximal normal subgroup of | #72312207, #72312219, #72312228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272331);