Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,12,9,24)(7,16,20,26,19,13,17,28)(8,21,23,29,22,14,25,30)(18,31,32,36,33,27,34,35);(1,2,3,4)(6,12)(7,17)(9,10)(13,26)(14,21)(16,28)(18,32)(19,20)(23,25)(27,35)(31,36)(33,34);(2,4)(5,7)(6,13)(8,18)(9,20)(10,17)(11,19)(12,16)(14,27)(15,26)(21,31)(22,33)(23,34)(24,28)(25,32)(29,35)(30,36);(5,8)(6,14)(7,18)(9,23)(10,25)(11,22)(12,21)(13,27)(15,29)(16,31)(17,32)(19,33)(20,34)(24,30)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312189, #72312228, #72312242, … |
| maximal normal subgroup of | #72312189, #72312228, #72312242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273833);