Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1254 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)(5,6,10,15)(7,16,21,27)(8,22,23,30)(9,13,17,14)(11,20,28,19)(12,24,31,25)(18,29,33,26)(32,35,36,34);(2,4)(5,7,17,20,10,21,9,19)(6,11,13,27,15,28,14,16)(8,18,25,35,23,33,24,34)(12,29,22,32,31,26,30,36);(5,8)(6,12)(7,18)(9,24)(10,23)(11,26)(13,22)(14,30)(15,31)(16,32)(17,25)(19,34)(20,35)(21,33)(27,36)(28,29);(2,4)(6,13,15,14)(8,23)(11,27,28,16)(12,30,31,22)(18,33)(24,25)(26,32,29,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,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,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321530 |
| maximal normal subgroup of | #72321530 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278207);