Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| 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 | |
2275 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)(7,16)(8,21)(9,25)(10,15)(11,19)(12,23)(13,22)(14,17)(18,27)(20,31)(24,30)(26,28)(29,36)(32,35)(33,34);(5,7,17,19,10,20,9,26)(6,11,14,31,15,28,25,16)(8,18,22,35,24,36,23,34)(12,29,21,33,13,27,30,32);(2,4)(5,8)(6,12)(7,18)(9,22)(10,24)(11,27)(13,15)(14,30)(16,32)(17,23)(19,34)(20,36)(21,25)(26,35)(28,29)(31,33);(5,8)(6,13)(7,18)(9,23)(10,24)(11,27)(12,15)(14,30)(16,33)(17,22)(19,35)(20,36)(21,25)(26,34)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,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^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287261, #72287270, #72287360, … |
| maximal normal subgroup of | #72287261, #72287270, #72287360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273930);