Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20)(23,29)(24,31)(26,32)(28,34)(30,35)(33,36);(6,10)(11,12)(13,15,16,19)(14,17,18,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(5,7,9,8)(6,11,10,12)(21,23,24,29,25,30,31,35)(22,26,28,32,27,33,34,36);(5,8,9,7)(6,12,10,11)(14,18)(17,20)(22,27)(26,33)(28,34)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240744, #72240762, #72240764, … |
| maximal normal subgroup of | #72240744, #72240762, #72240764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299739);