Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,22)(9,12)(10,15)(11,16)(13,21)(14,23)(18,27)(19,33)(20,28)(24,30)(25,29)(26,32)(31,34)(35,36);(6,12)(7,18)(9,10)(13,17)(14,22)(15,16)(19,32)(23,25)(26,31)(27,28)(29,30)(33,36);(5,7,10,21,11,18,9,20)(6,13,12,17,16,28,15,27)(8,19,23,34,24,32,25,35)(14,26,22,31,29,33,30,36);(5,8)(6,14)(7,19)(9,25)(10,23)(11,24)(12,22)(13,26)(15,30)(16,29)(17,31)(18,32)(20,35)(21,34)(27,36)(28,33)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312189, #72312228, #72312300, … |
| maximal normal subgroup of | #72312189, #72312228, #72312300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267779);